Statistics for Economics for Class 11 N. M. Shah
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statistics for economics for class 11...
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PREFACE TO THE THIRD REVISED EDITION Periodic updating and review accommodate new knowledge as well as adds freshness even as it allows for continuity. This third revised edition is an effort to that end. Statistics for Economics for class XI in its revised format brings forth the changed mode by the Central Board of Secondary Education, New Delhi in 2005-06. The revision includes sufficient exercises keeping in mind learning tools of Statistics in the context of the study of Economics. This volume incorporates extensive and colourful diagrams and illustrations to enhance a better and friendlier understanding of concepts of Statistics. Answer to the numerical questions in the exercise of unit 3 are also provided so that the students can verify the solutions. It is hoped that this revised edition will be of great help to both teachers and students. — N.M. SHAH PREFACE Stamtics has become an anchor for social, economic and scientific studies Stat,st,cal methods are widely used in several disciplines, be i, planning bul ' management, psephology (study of voting patterns), psycholo^ oT adve Sn; steps. A hst of formulae has been provided at the end of each chapter of unit 3 and rJr rTT —• ^ong years of teLh ng thrsubta I have to acknowledge that in the wriring of this volume I have got immense help rom my frtends and relation. The pubUshers have been very cooperative aTrelp ll 1 Trr r "tT-
-derstanding w'fe bes d s tsp rin '
me and looktng after the household, has done ind.spensable work for the volume Z M.M. Shah, himself a scholar and teacher of economics, and who retired as dean of acuity of Commerce, Nagpur University and Prmc.pal, G.S. College of clt rct Shri Ram College of Commerce, Delhi April, 2002 —N.M. SHAH SYLLABUS STATISTICS FOR ECONOMICS-XI One Paper
3 Hrs. 100 marks 104 Periods/50 xMarks 5 Periods/3 Marks 25 Periods/12 Marks 64 Periods/30 Marks 10 Periods/5 Marks PART-A STATISTICS FOR ECONOMICS 1.
Introduction
2.
Collection, Organisation and Presentation of Data
3.
Statistical Tools and Interpretation
4.
Developing Projects in Economics
Unit 1: Introduction What is Economics?
^^
Meanmg scope and importance of statistics m Economics Unit 2: Collection, Organisation and Presentation of Data p,,, SoToflot^tm^^^^^^^ W imrr^
-«
and National Sam;:;! sT^.y^lZZT
^^^^ ^^
Organisation of Data: Meaning and types of variables; Frequency Distribution. Unit 3: Statistical Tools and Interpretation interpretation for the rite d^ived)
P'-d'
deviat.o„,_Lore„z curvLMeaJ^aX ^^^ --Karl Pearson's method Oirrelation-meaning, scatter diagram-Measures of correlation-(two variables ungrouped data), Spearman's rank correlation "n^ inL'"on types-wholesale price index, consumer price "umLrs
production, uses of index numbers; inflation and iLx
Unit 4: Developing Projects in Economics Tu . J ■
Periods
in n • j
latlo^ both TaL tT^rV"
data, secondary
of L eimnl. f I' organisations outlets may also be encouraged. Some of the examples of the projects are as follows (they are not mandatory but suggestivT) (t) A report on demographic amongst households; suggestive). («) Consumer awareness amongst households (»/) Changing prices of a few vegetables in your market (w) Study of a cooperative institution—milk cooperatives ^eJ^t to enable the students to develop the ways and u1 CONTENTS UNIT 1 : Introduction 1.
What is Economics
2.
Introduction—Meaning and Scope .
UNIT 2 : Collection and Organisation of Data 3.
Collection of Primary and Secondary Data
4.
Organisation of Data
Presentation of Data 5.
Tabular Presentation
6.
Diagrammatic Presentation
7.
Graphic Presentation
UNIT 3 : Statistical Tools and Interpretation 8.
Measures of Central Tendency
9.
Positional Average and Partition Values
10.
Measures of Dispersion
11.
Measures of Correlation
12.
Introduction to Index Numbers
UNIT 4 : Developing Projects in Economics
13.
Preparation of a Project Report
1 10 22 52 76 87 108 138 178 232 313 354 394 WHAT IS STATISTICS? S:T:A:T:I:S:T:I:C:S: Scientific Methodology Theory of Figures Aggregate of Facts Tables and Calculation for Analysis Investigation Systematic Collection Tabulation and Organisation Interpretation Comparison Systematic Presentation tc fc ea dc in lif wl his in ( UNIT 1
—
HmODUCTIOr^ m Whui is ficonoinifs? ■ Jlieaiiiiig, Scope and Imporianct^ of l§)tatislics in Economms ■■
Chapter 1
what is economics? 1- Introduction 2.
Activity
3.
Definition of Economics
4.
Nature of Economics
Economics as a Science (i^ Economics as an Art introduction If each of us possessed 'Aladdin's magic lamp, which we had merely to rub in order to get our desires fulfilled immediately, there would be no economic problem and no need for a science of economics. In real life we are not lucky as Aladdin, we have to work to earn our livelihood. All people in this world work to satisfy their unlimited wants and desires. Every one requires food to eat, clothes to wear and house to live in. Besides these in daily life. They need television, mobile phone, motor bike, car etc., to lead a comfortable life. The person visits the market and enquires about the varieties and prices of the item which he wants to purchase. Thinking about his source and alternative choices, he uses his sense of economy and decides to buy that item. This is economics. So, A customer is a person who buys goods to satisfy his wants. A Producer is a person who produces or manufactures goods. A Service holder is a person who is in a job to earn either wages or salary to buy goods. A Service provider is a person who provides services to society to earn money, e g doctors, scooter drivers, lawyers, bankers, transporters, etc. All above persons are busy in different activities to earn, called economic activity in ordinary business of life. They face in their life the problem of scarcity of income z Statistics for Economics-XI »purcha. Thus, and^J^Ttu' "" ^r^'^ff'^^l^dge with economic activities relating to earning ar^ spending the wealth and tncome. Economics is the study of how human beinJZa^
tZ^ tn":^-' T"""
unlimited wLts in sZZ Z^LTZ
^omm maxnntse thetr satisfaction, producers can maximise their profits and society can maxtmtse its social welfare'. ^^ infn^W mT publication of Adam Smith's "An Inquiry into the Nature and Causes of Wealth of Nations", in the year 1776 At its Mxth Z name of economics was 'PoHtical Economy'. Some of the suggested names —
Catallactics or the science of exchange.
—
Plutology or the science of wealth.
—
Chrematistics or the science of money-making
~ "y't to^om- s''.'^'
-- ^
(HoItVoTr
has its origin into two Greek words : Oikos
E'^gli^h
'
'^o'^tical
(Household) and nomos (to manage). Thus, the word economics was used to mean home management with limited funds available in the most possible economLal mTn2 activity Iife.?herlf
"
^
day
1.
Non-economic Activities
2.
Economic Activities
or
Activities. These activities are those which have no economic aspect
or are not concerned with money or wealth, viz. ~ bkll^S ''
8«-together, attending
Dirtnday parties or marriages etc. " wSrCo?'''!'''^'"'^ ''
o,
&
^"^"dwara, mosque or church to
worship God, attending mass prayer (Satsang), etc. Political activities such as various activities performed by different political parties namely by Bhartiya Janata Party (BJP), Congress Party etc ~
^'ds or helping
-
Parental activities, such as love and affection towards their children
—^
^l^ey involve any
nf
Activities. Different types of activities are performed bV different types
of people (doctors, teachers, businessmen, industrialists, lawyers etc.) so as taelT^
What is Economics?
3
living. Every one is concerned with one or the other type of activity to earn money or wealth to meet their wants. An economic activity means that activity which is based on or related to the use of scarce resources for the satisfaction of human wants. Economic activities are classified as under : the ECONOMIC ACTIVITIES i
nt >ution kos ome I day ispect nding to lies ilping any [types I their Production : Production is that economic activity which is concerned with increasing the utility or value of goods and services. Manufacturing shirt with the help of cloth (raw material) and tailoring (labour) etc. is an act of production. Transporting sand from river bank to a town, where it is needed, is also an act of production. Here utility is created through transportation of goods to the person who needs it. Consumption : Consumption is that economic activity which is concerned with the use of goods and services for the direct satisfaction of individual and collective wants. Consumption activity is the base of all production activities. There would have been no production if there would have been no consumption. For example, eating bread, drinking water or milk, wearing shirt, services of lawyer or doctor etc. are consumption activities.
Investment : Investment is that economic activity which is concerned with production of capital goods for further production of goods and services. Investment indirectly satisfies human wants. For example, the production of printing press machines to print newspapers, books, magazines etc. or investment in computers to provide Internet, banking and related services. Exchange : Exchange is that economic activity which is concerned with sale and purchase of commodities. This buying and seUing is mostly done in terms of money or price. So, it is also called ""Product Pricing'' which relates to determination of the price of the product under different conditions of the market, viz., perfect competition, imperfect competition, monopoly etc. Distribution : Distribution is that economic activity which deals with determination of price of factors of production (land, labour, capital and enterprise). This is known as the 'Factor Pricing', e.g., price of land is rent, that of labour is wage, that of capital is interest and price of entrepreneur is profit. Distribution is the study to know how the national income or total income arising from what has been produced in the country (called Gross Domestic Product or GDP) is distributed through salaries, wages, profits and interest. "Economics is that branch of knowledge that studies consumption, production, exchange and distribution of wealth". —Chapman 10
Statistics for Economics-XI
definition of economics Economics has been defined by many economists m different ways The set of cat"go^s^'"'" '' ^^ mto the folwL; W 1.
Wealth definition—Adam Smith
2.
Material welfare definition—Alfred Marshall
3.
Scarcity definition—Lionel Robbins
4.
GroAvth definition—Paul A. Samuelson
1. Wealth Definition (/) Adam Sm^th the father of modem economics, m his book 'An Inquiry mto the Nature and Causes of Wealth of Nations' in 1976 defined thatproduction and expansion of wealth as the subject matter of («) According to J.B. Say, Economics as "the science which deals wtth wealth"
ofTaTth!
"
^^^ consumption
{in) Ricardo shifted the emphasis from production of wealth to distribution of wealth Criticism : This definition is not a precise definition. It gives importance to wealth rather than production of human and social welfare. importance to wealth The wealth definition of economics was discarded towards the end of the 19th century. 2. Material Welfare Definition "Economics is a sUuiy of mankind i„ the ordinary business of life it examines that What is Economics? 5 Criticism : {a) In economics, we study immaterial things also. (b)
Welfare cannot be measured in terms of money.
(c)
Welfare definition makes economics a purely social science.
II h "Tf
""
"
- d^ff-nt times.
Then basic difference between Adam Smith's and Marshall's definition is that Ad.m r-iti* sr. ™ -—hiri^3. Scarcity Definition There are three important aspects in this definition. They are • Icf of Tir m" human wants which is the tact of Me. When one warn gets satisfied, another want crops up. fr"
''
are scarce in relation
coal IS used m factories, m running railways and in thermal stations for electric generation and by households, etc. electric In short, according to Robbins, Economics is a science of choice It deals with how Crit^sm : Sfet™s Scarcity definition of economics has been criticised on the following grounds ■ (0 The defimtion is impractical and difficult. It is narrow and restricted in scope It ^ development. It has notS^^^
Hi) The definition makes economics a human science. 4. Growth Definition Paul A. Samuelson defines— to p^u^ vanous commod,Ues overtime and distribute them for consumptionZ^ or 12
Statistics for Economics-XI
The definition combines the essential elements of the definitions by Marshall and Robbms. Accordmgly, economics is concerned with the efficient allocation and use of scarce means as a result of which economic growth is increased and social welfare is promoted. The definition has been accepted universally. In short, the growth definition of economics is most comprehensive of all the earUer definitions. iture of economics ^ Nature of economics—as a science or art. It is science and art as well. NATURE OF ECONOMICS ics as Art A. ECONOMICS AS A SCIENCE Science can be divided into : (a) Natural science, and (b) Social science : Sciences like Physics Biology and Chemistry are natural or physical sciences, where experiments can be conducted in the laboratory under controlled conditions. Relationships can be decided between cause and effect, which are based on facts. Observations can be made and used to prove or disprove theories. The results apply universally. Economics is a social science because it is systematic study of economic activities of human beings. Economics is a science as it is a branch of knowledge where various facts have been systematically collected, classified and analysed. The following arguments are given in favour of economics as a science. (/) Systematised Study : The study of economics is systematically divided into consumption, production, exchange and distribution of wealth and finance which have their own laws and theories. Economics as social science which is a systematic study of human behaviour concentrating on maximum satisfaction to households maximum profit to producers and maximum social welfare to the society as a whole. ^
Hi) Scientific Laws : Economics is a science because its laws are universally true Different laws m economics namely, law of demand, law of supply, law of dimimshing marginal utility, law of returns, Gresham's law etc. are applicable to all types of economies, whether capitaKstic, socialistic or mixed economy y ae. of ; to What is Economics?
y
(m) Cause and Effect Relationship : Economic laws establish cause and effect relationship like the laws in other sciences. For example, the law of demand shows the relationship between change in price and change in demand..It shows that mcrease in price of a commodity (the cause) will decrease its demand (the effect) establishing the negative or inverse relationship between price and quantity demanded. The law of supply shows that the increase in price of a commodity (cause) will increase its supply (the effect) establishing the positive relationship between price and supply of quantity of commodity. (iv) Verification of Laws : Like other sciences economic laws are also open to verification. These economic laws can be verified through any empirical investigation. On the basis of the arguments given above, we can say economics is a science—not exacriy natural or physical science but social science that studies economic problems and policies in a scientific manner. Economics—A Positive or Normative Science (a) Economics as a Positive Science A positive science is one which makes a real description of an activity. It only answers what ts} what was! It has nothing to suggest about facts, positive economics deals with what IS or how the economic problems facing a society are actually solved. Prof. Robbins held that economics was purely a positive science. According to him, economics should be neutral or silent between ends; /.e., there should be no desire to learn about ethics of economic decisions. Thus, in positive economics we study human decisions as facts which can be verified with actual data. Some exampi es of Economics as a positive science are : {i) India is second largest populated country of the world. (k) Prices have been rising in India. (m) Increase in real per capita income increases the standard of living of people. (iv) The targeted growth rate of the tenth five-year plan is 8 per cent per annum. {v) Fall in the price of commodity leads to rise in its quantity demanded.
(vi)
Minimum wage law increases unemployment.
(vii) The share of the primary sectors in the national income of India has been declining. {viii) Ordinary business of life is affected enormously by tsunami, earthquakes, the bird flue, droughts, etc. (b) Economics as a Normative Science A normative science is that science which refers to what ought to be} what ought to have happened} Normative economics deals with what ought to be or how the economic problems should be solved. Alfred Marshall and Pigou have considered the normative aspect of economics, as it prescribes that cause of action which is desirable and necessary to achieve social goals. It makes an assessment of an activity and offers suggestions for that. The statements which make assessment of activity and offer suggestions are called 14
Statistics for Economics-XI
normative statements. The normative statements, in fact, are the opinions of different persons relating to tightness or wrongness of a particular thing or policy. Normative statements cannot be empirically verified. That part of economics which deals with normative statements is called Normative Economics. Thus, economics is both positive and normative science. F^smvc Some examples of Economics as a normative science are : (/) Minimum wages should be guaranteed by the government in all economic activities. (//) India should not take loans from foreign countries. [Hi) Rich people should be taxed more. [iv) Free education should be given to the poors. {v) Effective steps should be taken to reduce income-inequalities in India. (vi) India should spend more money on defence. {vii) Government should stop minimum support price to the farmers. (vtii) Our education system should produce sufficient qualified and trained persons to the economy. Economics as positive science and normative science is inseparable. In reality economics has developed along, both positive and normative lines. The role of economist is not only to explain and explore as positive aspect but also to admire and condemn as negative aspect which is essential for healthy and rapid growth of economy.
In the followii^ examples first part of statement is positive giving facts and second i part IS normative based on value judgements. H) Indian economy is a developing economy, the government should make development through correct and proper planning. (ii) A rise in the price of a commodity leads to a fall in demand of quantity of commodity, therefore government should check rise in prices. (iti) Rent Control Act provides accommodation to the needy peoples, therefore, the act should be honestly implemented. B. ECONOMICS AS AN ART Art IS practical application of knowledge for achieving some definite aim. It helps in solution of practical problems Art is the practical application of scientific principles. Sc ence lays down princip es while art puts these principles into practice. Economics is an art as it gives us practical guidance in solution to various economic problems. ' We all know that there is oil shortage in India. The information given by economics .sposmve sconce We also know the govermnent aims at removing' oil shortage X information supplied by economics is normative science. In order to achieve the objective of full availability of oil m India, the govermnent has followed the path of oil plaLng The path of planmng is an art as it implies practical application of knowledge with a view to achieve some specific objectives. So, we can say that economics is an art. Economics is, thus, a science as well as an art. What is Economics?
y
exercises i Explain the origin of word 'Economics', i What is economic activity? Distinguish between non-economic and economic activities. Make a Ust of economic activities that constitute the ordinary business of life. What are your reasons for studying Economics? How will you choose the wants to be satisfied? Give Adam Smith's definition of economics. Define economics in the words of Alfred Marshall. "Economics is the science of choice." Explain. Which is the most accepted definition of economics? Give the definition. Explain welfare definition of economics.
"Economics is about making choices in the presence of scarcity." Explain. How scarcity and choice go together? What is meant by economics? Economics is a science? Give reasons. Discuss the nature of economics as a science. Give argimient in favour of economics as a science. Is economics an art? Give reasons. Is Economics a science or an art? Explain with reasons. Is economics a positive science or a normative science or both? Explain. lin 3. 4. 5. 6. Chapter 2 introduction-meaning and scope Introduction What is Statistics? Functions of Statistics Importance of Statistics Limitations of Statistics Misuse of Statistics i introduction e—,
plannW and X
^^^^ ^^ ^
progJsst tcTs - - developments and chemistry, medicine, technology etc) neLTnWr' new machines have been devdoS'tLt I f ff because man-whether Indtn
' ^^"^'^es of energy,
T? -^f^^ble. All this is possible
thinking and reasoning which had evo^ given us civiIisation4he wtef
^P^^^^^es,
'""I' ^^ ^^"^^^e-is gifted with
f ^^ has
electricity, machinery etc. We ntw ^^ ^^e irrigation system, systems, better organisations for the comX bul^^^^^^^ All thic h^c , "-"'npiex Dusiness and administration today
wayfoftalSmnrp^^^^^^^
^PP^'ed itself in'findmg
and scientific man'ner. A meZdo^X b'n consists Jf la^^^^
^ things. The empirical methodolog^
mformation, analysing the information Th^ ^
observations and collecting
conclusions by fu^er^bserta^TsZs W al ZtZ' ^^^^ ^ knows it or not, he uses this method tof'""'.^'e made Whether a common man buying vegetabks he looks t dS^lr^u^^^^^^^^^^
decision-makmg. While
and then mentally calculates, or ^rks out wLT.
^^ops
P."'''
observes from his daily ex^elre whT'> ^^^ ^hich shop. A shopkeeper decides to stock these
- demand, a'nd
the pattern of demand and manufactures larle o^T "^^^ufacturer also observes or manufactures new items acco^SnSr^rthTZL^
^^^ demand
radio, the television etc., it is possiWe trcXr T f "^^^^"^edia-the newspaper, the Introduction—Meaning and Scope \ 11 demand and supply, he collects data (information) systematically, gets it organised in some logical or systematic way, analyses this data according to certain principles and draws conclusions. He has to do it carefully since a wrong judgement can completely ruin him. Quantitative Data and Qualitative Data : An empirical investigation is an investigation where facts are collected through observation. In Physics, Chemistry and Botany, only those things that can be observed by our senses—seeing, hearing, touching, tasting and smelling—are taken to be reliable and then recorded (noted). We all agree that the rose is beautiful. How do we reach that conclusion? We all like its colour, shape and above all its smell. In this respect, it is not a subjective or personal conclusion. But I say that I like the rose most of all the flowers, this would be a subjective statement. A scientist however, makes very precise statement—he would say that roses have a sweet smell. Similarly, people might say that theft and robbery have increased these days. This might be a conclusion based on impression people get from the newspaper reports of cases of theft and robbery. This impression may or may not be true. We can find out whether it is true or not only by comparing the number of cases of theft and robbery reported during one year with
the number of cases reported in other years. An investigator would collect such information from police records. When information or observations are recorded in numbers or quantity, we say we have quantified information. For example, the number of people in a state who are strict vegetarians, heights or weights of students, everyday temperature, income of individuals, prices of wheat during this week, number of people in country are really poor-rich-middle class, number of people are illiterate who will not get jobs, number of highly educated and will have best job opportunities, etc. are known as 'Quantitative data'. However, not all information can be numerically expressed. It is not possible in certain cases to measure or quantify information, e.g., preference of people viewing TV. channels, intelligence of students, appreciation of art, beauty, music etc. Supposing a selection for a post is to be made, candidates are interviewed, some questions are put to them and their qualifications are taken into consideration. The interview board discusses the comparative merit of the candidates and ranks them for final selection. This judgement is not quantifiable, it is based on impression. Non-quantifiable/qualitative items can however be measured in percentages. For example, percentage of people watching TV. news in English or Hindi or other regional languages. This information obtained in percentages is called 'Qualitative data'. It may be collected through questionnaire or opinion poll using landline or mobile telephone, internet or newspapers. Social sciences, such as economics, sociology, management etc., do not always deal with what we call inherently measurable or quantifiable facts. is smistics ? > It is necessary to have quantitative measurements even for things which are not basically quantifiable. This is necessary for preciseness of statement. The systematic h 12 vw 11 ^^'^tistics for Economics-Xl treatment of quantitative expression is known as 'Statistics'. Not all quantitative expressions are statistics; we will see that certain conditions must be fulfilled for a quantitative statement to be called statistics. We will also consider later the functions and hmitations of statistics. First, let us understand what comes under the name Statistics. Statistics can be defined in two ways : (a)
In plural sense.
(b)
In singular sense.
i^nn'W^nir"*
'consider whether figures
1600, 400, 80, 20, 700, 300, 70 and 30 are Statistics? Figures are innocent and do not speak anythmg. But when they refer to some place, person, time etc., they are called statistics. Let us look at the table given below : Students in Two Schools (2005-2006) Kendria Vidyalaya
Govt. Senior Secondary School
Students
Number
Percentage Number
Boys Girls
1600 400
80 20 700 300
Total 2000 100
Percentage
70 30
1000 100
The above table gives a numerical description of students in Kendria Vidyalaya and Govt Senior Secondary School. Students are grouped as boys and girls and percentage is st^ calculated for each group. Now, in this context the figures 1600, 400, 700 etc have a ' of statistical meaning; we call this statistics of students. Similarly, we find in newspapers ^ statistics of scores in a cricket match, statistics of price, statistics of agricultural production, i sin statistics of export and import etc. j ^ " martT^^'^Vl
Statistics we mean aggregates of facts affected to ^ ^
marked extent by muhtphctty of causes numerically expressed, enumerated or estimated according to reasonable standards of accuracy, collected in a systematic manner for a predetermmed purpose and placed in relation to each other." The above definition covers the following main points about statistics as numerical presentation of facts (Plural sense). Statistics are aggregates of facts : A single observation is not statistics, it is a group of observations, e.g., "pocket expenses of Anil during a month is Rs 50" is not statistics. But "pocket expenses of Anil, Prakash, Sunil and Suresh during a month are Rs 50, 55, 80 and 70 respectively" are statistics.
{b) Statistics are affected to a marked extent by multiplicity of causes : Statistics are generally not isolated facts they are dependant on, or influenced by a number of phenomena, e.g., electricity bills are affected by consumption and rate of electricity (c) Statistics are numencally expressed : Qualitative statements are not statistics unless A they are supported by numbers. For example, if we say that the students of a class colle. Introduction—Meaning and Scope 13 are very good in studies, it is not a statistical statement. But when a statement reads as 40 students got first division, 30 second division, 20 third division and 10 failed out of 100 students, it is a statistical statement expressed numerically. (d) Statistics are enumerated or estimated according to reasonable standard of accuracy: Enumeration means a precise and accurate numerical statement. But sometimes, where the area of statistical enquiry is large, accurate enumeration may not be possible. In such cases, experts make estimations on the basis of whatever data is available. The degree of accuracy of estimates depends on the nature of enquiry. (e) Statistics are collected in a systematic manner : Statistics collected without any order and system are unreliable and inaccurate. They must be collected in a systematic manner. if) Statistics are collected for a pre-determined purpose : Unless statistics are collected for a specific purpose they would be more or less useless. For example, if we want to collect statistics of agricultural production, we must decide before hand the regions, commodities and periods for which they are required. (g) Statistics are placed in relation to each other : Statistical data J»re often required for comparisons. Therefore, they should be comparable periodwii>c, regionwise, commoditywise etc. When the above characteristics are not present numerical data cannot be called statistics. Thus, "all statistics are numerical statements of facts bui all numerical statements of facts are not statistics." Statistics defined in singular sense (as a statistical method) : Statistics in its second, singular sense, refers to the methods adopted for scientific empirical studies. Whenever a large amount of numerical data are collected, there arises a need to organise, present, analyse and interpret them. Statistical methods deal with these stages : PRESENTATION
I Interpretation Statistics as Methodology According to Croxton and Cowden, "Statistics may be defined as a science of collection, presentation, analysis and interpretation of numerical data.'" It
Statistics for Economics-XI The above definition covers the following statistical tools : (a) Collection of data : This is the first step in a statistical study and is the foundation of statistical analysis. Therefore, data should be gathered with maximum care by the investigator himself or obtained from reliable pubHshed or unpublished sources (b) Organisation of data : Figures that are collected by an investigator need to be organised by editing, classifying and tabulating. (c) Presentation of data : Data collected and organised are presented in some systematic manner to make statistical analysis easien The organised data can be presented : with the help of tables, graphs, diagrams etc. (d) are
Analysis of data : The next stage is the analysis of the presented data. There
large number of methods used for analy sing the data such as averages, dispersion correlation etc. ' (e) Interpretation of data : Interpretation of data implies the drawing of conclusions on the basis of the data analysed in the earlier stage. On the basis of this conclusion certain decisions can be taken. Stages of Statistical Study According to the figure, interpretation of data is the last stage in order to draw some conclusion. One has to go through the four stages to arrive at the final stage; they are — collection, organisation, presentation and analysis. First stage — collection of data refers to gather some statistical facts by different methods. The second stage is to organise the data so that collected information is easily intelligible. This is the arrangement of data in a systematic order after editing. Third stage of statistical study is presentation of data After collection and organisation the data are to be reproduced by various
niethods of presentation, namely tables, graphs, diagrams, etc. so that different characteristics of data can easily be understood on the basis of their quality and uniformity. Fourth stage of statistical study is the analysis of data. Calculation of a value by different methods and tools for various purposes is made to arrive at the last stage of study viz interpretation of data. ^ In brief statistics is a method of taking decisions on the basis of numerical data properly collected, organised, presented, analysed and interpreted. in i scie drai relai ((
functions of statistics Following are the functions of statistics : 1. Statistics simplifies complex data : With the help of statistical methods a mass of data can be presented in such a manner that they become easy to understand. For example, the complex data may be presented as totals, averages, percentages etc Stati I resea Mars quan; of stj Introduction—Meaning and Scope
\5
2. Statistics presents the facts in a definite form : This definiteness is achieved by stating conclusions in a numerical or quantitative form. 3. Statistics provides a technique of comparison : Comparison is an important function of statistics. For example, comparison of data of different regions; periods, conditions etc., is helpful for drawing economic conclusions. Some of the statistical tools like averages, ratios, percentages etc., are used for comparison. 4. Statistics studies relationship : Correlation analysis is used to discover functional relationship between different phenomena, for example, relationship between supply and demand, relationship between sugarcane prices and sugar,
relationship between advertisement and sale. Statistics help in finding the association between two or more attributes, for example, association between literacy and unemployment, association between innoculation and infection etc. 5. Statistics helps in formulating policies : Many policies such as that of import, export, wages, production etc., are formed on the basis of statistics. Some laws such as Malthus' theorj^ of population and Engel's law of family expenditure are based on statistics. 6. Statistics helps in forecasting : Statistics also helps to predict the future behaviour of phenomena such as market situation for the future is predict^).' on the basis of available statistics of past and present. Economist might be interested in predicting the changes in one economic factor due to the changes ir another factor. For example, he might be interested to know the impact of today's investment on the national income in future which is possible with the knowledge of statistics. 7. Statistics helps to test and formulate theories : When some theory is to be tested, statistical data and techniques are useful. For example, whether cigarette smoking causes cancer; whether demand increase affects the price, can be tested by collecting and comparing the relevant data. importance of statistics The use of statistical method is so widespread that it has become a very important tool in affairs of the world. It IS indispensable to fields of investigations especially in the sciences, such as Botany, Sociology, Economics, Medicine etc. It helps particularly in drawing research conclusions. Let us examine the importance of statistics in some fields relating to economics and business : {a) Statistics and Economics
(b) Statistics and Economic Planning
(c) Statistics and Business
(d) Statistics and Government
Statistics and Economics A number of economists have given a practical shape to statistical tools for economic research. Famous economists (like Augustin, Cournot, Vilfredo Pareto, Leon Walras, Alfred Marshall, Edgeworth, A.L. Bowley etc.) evolved a number of economic laws by quantitative and mathematical studies^ In India, Prof. P.C. Mahalanobis, Dr V.K R V Rao R.C. Dcsai, ctc. iiave cgntrmuieu aTOtWtne aeveiopmeiu ui lucuiclk-^t-----' of statistics. !H
If 16 Statistics for Economics-XI ...ril—T
tools and the importance of
ZnomV: rr' f" "T™" ^^^^^ - ^ relanonLp amo^ Tecorm^rA ?
of mathematics andLtistic!
m economics. As a result, a new science has evolved which is called Econometrics the Z" ^r'
have evolved due to statistical analysis in
the f eld of economics, Engel's law of family expenditure, Malthus theory of population etc. New things are being invented today in all the sciences becauTo7the econoT^b^""" T^
• T ^^^^
^
new laws in
Statistical methods have made a contribution to the
development of empirical side of economics; the inductive method of economics is dependent upon statistical methods. economics is TT''
are the tools and
appliances of his laboratory, m the same way as the doctor uses stethoscope for diagnosis of a patient. A number of economic problems can easily be understood by the useTf tatistical tools. It helps m formulation of economic policies.^ Let us unLstand the importance of statistics keeping in view the various parts of economics oTrh!" consumption : Every individual needs a certain number 7 K'
necessities, then on comforts and luxuries, which
depend on his income; but there is no end to his desires and demands. No sooner does he consume one thing, he desires to obtain the other. We discover how staS^" T"^' consumption. The Td/vT r
T rt^ ^^e taxaWe liability of
individuals and their standard of living
(b) Statistics and the study of production : The progress of production every year can easily be measured by statistics. The comparative study of the prod^Lty of various elements of production (e.g., land, labour, capital and enterprise) is also done with the help of statistics. The statistics of production are ver^ helpful or ad^stment of demand and supply. Every developed country executes L census of production with a view to make a comparative study of various fields of production and economic planning. ^ wuucuou fn^ttirf
=
^^
- -^onal and
international demand. A producer needs statistics for deciding the cost of . P^r ^^^^ ^^
competition afd demand o
^mmodity m a market. The law of price determination and cost price which are : (d) Statistics and the study of distribution : Statistics are helpful in calculation of national income in the field of distribution. Statistical methods are used in solving the problem of the dismbution of national income. Various problems arise di^I to o7lttfc7datr "
^^ ^ with"hrhelp
et/eo- other econormst, have to make bricks." -Marshall Introduction—Meaning and Scope
1^
Thus, statistics is useful in the various fields of economics. It gives statement of facts, direction to solve problems, evolution of economic laws and helps in economic planning. Economic laws in the modern economic world are based on mathematics and statistics which help to form econometric models; these models are helpful in solving economic problems. On this basis we can call economics a Science of Human welfare and statistics as an Arithmetic of Human welfare. Statistics and Economic Planning Statistics is the most important tool in economic planning. Economic planning is the best use of national resources, both human and natural. Planning without statistics is a leap in the dark. Every phase in planning—drawing a plan, execution and review is based on statistics. The success of a plan is dependent upon sufficient and accurate statistical data available at all these stages. The comparison of the stage of development of one country with other is possible only with the availability of statistical data. There are a number of problems of
underdeveloped countries, e.g., over population, lack of industries, lack of agricukural development, lack of education etc. These problems can be fully viewed and understood only by getting the actual figures for different areas. Similarly, general review of progress in all fields of economic development needs the help of statistical data and statistical methods. Priorities of expenditure of a national budget can be determined through the comparative study of past performances with the present. Thus, planning without statistics is a ship without radar and compass. Statistics and Business Planning Business activities can be classified as under : f" L BUSINESS 1 I Internal Wholesale Retail International Import Export i of Trade ^-1-1 t
V
Banking Transport Insurance Wterehousing Packmg Advertisement Mf
18 vw 11 ^^'^tistics for Economics-Xl of statmical method have to be followed The it the producer to f„ the pnces of ^o^rd""'
steps
method, of .ta„st,ea. profttable trade he must know what the ZoZ, '' would last. This is very .mportant or
demand
e„po„ for var.o„s co.mod,t.es and
at a^d
celling activities. For
he has to forecast when the dLaXodd Ll": °nf "Tf of reserves he must have. Similarly he ^st
t
^-We what amounts
h.s deposttors, otherwise his bak wol fT^oT rh transaction ,s required where statistical toTls tt iX-b^^
i
hfe e^rrtt^r that is ! what proportion of the. capS
decde' i
payments of matured policies
^ Proportion kept ready for
n>.ght fix for the same. In fact no mod^Xanirr °°
they
without analysis of the complex faJ^rthrLr"" ^ business analysis statistical tools are afeolt; Tfential analys^p tz: p":^-lt:tsere 717
^ ^^ ^
Statistical tools of collection, classification L uncertainties, data are used in all ma,or functioTstfinterpretation of material control, budgetary control, fmanc aTclro^f T"'' " ^nd so on.
'
control, cost control, personnel management
Statistics and Government Introduction—Meaning and Scope
\9
like that of crimes, taxes, wealth, trade etc., so that there is no obstacle for the promotion of economic development. Such policies may develop the economic status of the people and the nation, depending upon the accuracy of the statistical law. Statistics is indispensable for all important functions of the ministries of the state. Above all the ministry of planning takes into account statistics of various fields of economy. Keeping in view the increase of global price rise, the ministry plans and makes policy to import oil in 2010, which depends on expected oil production by domestic sources and likely demand for oil for the year 2010. The
policy of family planning can be made effective in controlling the population of country. Thus, statistical techniques are used to analyse economic problems of country, viz., unemployment, poverty disinvestment, price control, etc. Sometimes to make plans and policies, planners require the knowledge of future trend. This trend could be based on the data of past years or recent years. The required data can be obtained by surveys. For example, production poHcy of 2010 depends on the consumption recorded in past years and recent years which decides the expected level of consumption in 2010. This helps the planners to make the production policy for the future. The Ministry of Finance is responsible for preparing the annual budget of the country for which reliable statistical data of revenue and expenditure is necessary. In short, statistical tools are of maximum utility in the governance of state and formulation of various economic policies. f'i umitations of statistics Statistics is very widely used in all sciences but it is not without limitations. It is necessary to know the misuses and limitations of statistics. The following are the limitations of statistics. 1. It does not study the qualitative aspect of a problem : The most important condition of statistical study is that the subject of investigation and inquiry should be capable of being quantitatively measured. QuaHtative phenomena, e.g., honesty, intelligence, poverty, etc., cannot be studied in statistics unless these attributes are expressed in terms of numerals. 2. It does not study individuals : Statistics is the study of mass data and deals with aggregates of facts which are ultimately reduced to a single value for analysis. Individual values of the observation have no specific importance. For example, the income of a family is, say Rs 1,000, does not convey statistical meaning while the average income of 100 families say Rs 400, is a statistical statement. 3. Statistical laws are true only on an average : Laws of statistics are not universally applicable Hke the laws of chemistry, physics and mathematics. They are true on an average because the results are affected by a large number of causes. The ultimate results obtained by statistical analysis are true under certain circumstances only. 4. Statistics can be misused : Statistics is liable to be misused. The results obtained can be manipulated according to one's own interests and such manipulated results can mislead the community.
20
h (Si. ^ • , . ^t^tistics for Economics-XI observations of mass data nXkrT L
^ased on
to rectify them. Therefor^ it^eslnreT" " -...ca, states are a fa/.re m ^ misuse of statistics statistics by deliberately twisting or man pulat^ne^
^^ ^--e
be mterpreted by a lawyer to prove ^^ZTltTl^^;' " " ^^^ ^^^ tools for grinding their owTa^sf b^ student of commerce and economic , hou^ 1 the This generally takes place at the time of sefectinZm^^^^^ a and interpreting analysis of data^
"^^^ing comparisons
and^^eS-pJT~ - support knowledge of statistics, the truth with the help of his exercises
1- D.stm8„,sh between qualitatrve and quantitative data 3
Dto Tr-
4
M rr
-
above
characteristics?
and plural sense
obrva'Sn'" cr™"'"
-d never with a single
llT"" '' counting." Discuss. ■ ............. 4. 5. 6.
7. 8. Introduction—Meaning and Scope 21 16. 17. 18. 19. 20, Discuss with illustration the importance of Statistics in the solution of social and economic problems. "Statistical Analysis is of vital importance for successful businessmen, economists, administrators and educationists." Discuss with illustrations. Write notes on : (a) Importance of statistics in modern economic set up, {b) Statistics in economic analysis. Define Statistics. Explain its utiHty in the field of economic planning. "Statistical thinking is as necessary for efficient citizenship as the ability to read and write." Explain this statement in about 200 words. "Statistics in these days is indispensable for dealing with socio-economic problems". How far is this statement true? What is the importance of Statistics in modern economic set up? Explain giving examples.
. i- u
"Planning without Statistics is a ship without radar and compass." In the light ot this statement explain the importance of Statistics as an effective aid to national planning. Explain the relationship between Economics and Statistics and discuss how far it is correct to say that the science of economics is becoming statistical in its method. Explain briefly : (a) Statistics,
(b) Statistical methods,
(c) Statistical data, (d) Statistics in economic analysis.
Statistical methods are no substitute for common sense, comment. "The Government and poHcy maker use statistical data to formulate suitable policies of economic development". Illustrate with two examples. Mark the following statements as true or false. J (i) Statistics is of no use to economics without data. (ii) Statistics can only deal with quantitative data. (Hi) Statistics solves Economic problems. UNIT 2 }1: If K ^lecton and organisation of data 12 « CoUecAion of Primary and S p Organisation of Uata « Presentation of Data Chapter 3 collection of primary and secondary data What is a Statistical Enquiry? Sources of Data g Primary and Secondary Data Drafting the Questionnaire Methods of Collecting Primary Data Census and Sample Surveys Sample Surveys Methods of Sampling Random Sampling Non-Random Sampling Advantages of Sampling Reliability of Sample Data How Secondary Data is Collected? Some Important Sources of Secondary Data Census of India National Sample Survey Organisation (NSSO) COI cer are coIJ Sourct w other, statisti s-XI
Collection of Primary and Secondary Data
what is a statistical enquiry ?
Enquiry means a search for truth, knowledge or information. Statistical enquiry therefore means a search conducted by statistical methods. There are different subjects on this earth; some are described by the degree of expression (quality) and
some by the degree of figures or magnitudes (quantity). The application of a statistical technique is possible when the questions are answerable in figures (quantity), in other words the first and the foremost condition for the answer to the questions in statistical enquiry should be quantitative, for instance : Profit of firms measured in rupees; Income of families measured in rupees; Weight of students measured in kg; Age of students measured in years; Intelligence measured in marks obtained by students in a particular test. But, there are questions like—How great was Jawaharlal Nehru? How brave was Bhagat Singh? etc., which cannot be answered through statistical methods. Questions that can be answered in quantity lies within the purview of statistics, viz.. What is the average production of rice per acre in India? What is the total population of India? How many students are there in a class? Thus, statistical enquiry means statistical investigation or statistical survey, one who conducts this type of enquiry is called an investigator. The investigator needs the help of certain persons to collect information, they are known as enumerators, and respondents are those from whom the statistical information is collected. Survey is a method of collecting information from individuals. Let us observe the following table. TABLE 1 Production of Finished Steel in India (in Million Tons) fc^ year
Production
1950-51
1.0
1980-81
6.8
1990-91
13.5
2000-01
30.3
2001-02
31.1
2002-03
34.5
2003-04
36.9
Source : Government of India, Economic Survey 2004-05. We observe that the production of finished steel in India, is different from one year to other. They are not same. They varies from year to year. They are called Variables in statistics which is represented as X, Y or Z variables. The finished steel production in ti i-. 24
,
Statistics for Economics-XI V V tu f
by x-variable and the production of finished
From the followmg text, we will understand : 1.
What is the source of data?
2.
How do we collect data?
3.
By which method of survey is data collected
. sources of data BefoT^n' Z" ^^
'^^ta. This is the first stage in statistics!
SOURCES OF DATA
dary ^
Governmern departments RaiZr^cf
'
^^
^reparmg^at •
cantr^ine??""'"'"
external data whij
^^ary and secondary
'
Collection of Primary and Secondary Data
25
But, you may have the other choice that of visiting the factory accounts department, and record the information from the salary register or, may gather this information from the published report of the factory about the payment of wages. This is secondary source for an investigator but, for the factory it is a primary source.
Thus, primary data is collected originally and secondary data is collected through other sources. Primary data is first hand information for a particular statistical enquiry while the same data is second hand information for an another enquiry. The same data is primary in one hand and secondary in the other, e.g., any Government publication is first hand (Primary) for Government and second hand (Secondary) for a research worker. Thus, secondary data can be obtained either from published sources or from any other source, for example, a website which saves time and cost. PIHMARY DATA-* PUBLISHED How Primary Data is Collected The most popular and common tool is questionnaire/interview schedule to collect the primary data. The questionnaire is managed by the enumerator; researchers or trained, investigators. Sometimes the questionnaire is managed by the respondents also. MIMTim Following are the basic principles of drafting questionnaire : (1) Covering letter : The person conducting the survey must introduce himself and make the aims and objectives of the enquiry clear to the informant. A personal letter can be enclosed indicating the purposes and aims of enquiry. The informant should be taken into confidence. He should be assured that his answers will be kept confidential and he will not be solicited after he fills up the questionnaire. A selfaddressed and stamped envelope should be enclosed for the convenience of the informant to return the questionnaire. (2) Number of questions : The informant should be made comfortable by asking minimum number of questions based on the objectives and scope of enquiry. More the number of questions, lesser the possibiUty of response. Therefore, normally \l (I ■• I-J Ni m.26 Statistics for Economics-XI .nstr„c,.ons about units of measutement shol b^ g.7en
questionnaire, |
investigator. numbered for the convenience of the informant and the ''' sraetsr^s r— They the mformant should ^ abrtr^ve Ae aZ^f "" 'TT' ■"."''ject.ve. For this the blank space, e.g., ® ^y "smg a tick-mark in WUch of the folloJng languages you use most for uniting, (Pu, a cross) (.) English p M Punjabi
□
iit,)Vrd>x
n
(f) Any other q Sd fnTu^f: 'ir^'s'ts^;^:' ^^^-omd be ■Wrong', e.g..
are answerable m'Yes' or 'No'
or 'Right' or Are you married?
Yes/No
Are you employed? Yes/No should start from general These questionsleXle aS In which class do you read? In which subject you are more interested? 'rr
^-ooses. such
questi-ons should bet Smlt ^^l^rrt^^ ^^^^^^^^^^ Collection of Primary and Secondary Data A SPECIMEN QUESTIONNAIRE 27 H,. S.I Hit: 28 1 Statistics for Economics-xi Example :
work. Such
ml" "" "
"f P"- ^'"dents in Universi^-
W How will you solve the wage problem in your mdustry? (a)
Which brand of tea do you take?
(b)
Why do you prefer it?
(16)
T .ssn'r btk"
(b)
Do you love your children?
(c)
Do you beat your wife?
t
iiethods of collectiiig priiiiiary data etc. enq in a not Mer 7. 8. We^wmg are the methods of primary data collection which a« in common use \ Collection of Primary and Secondary Data COLLECTION OF DAW of 29 r lARY SECONDARY ►Direct Personal Interview ►Indirect Personal Interview ►Telephone Interview •^Information from Correspondents ►Mailed Questionnaires ►Questionnaires Filled by Enumerators 1 Published Sources
-1 Unpublished Bourns »Government Publications ► Publications of Internal Bodies ►Semi-official Publications Report of Committees and Commissions —> Private Publications (a)
Journals and Newspapers
(b)
Research Institutions
(c)
Professional Trade Bodies
(d)
Annua! Reports of Joint Stock Companies
(e)
Articles, Market Reviews and Reports
etc and collect the desired information. In the same way one can think of personal Imryof collection of information regarding family budget and living conto Ta group area. The investigator must be skilled, tactful, accurate, pleasing and should not be biased. Merits : 1.
Original data are collected by this method.
2.
There is uniformity in collection of data.
3
The required information can be properly obtained.
4
There is flexibility in the enquiry as the investigator is personally present.
5" Information can be obtained easily from the informants by a personal interview. 6. Since the enquiry is intensive and m person, the results obtained are normally reliable and accurate. 7 Informants' reactions to questions can be properly studied. ,,., 8'. Investigators can use the language of communication according to the educational standard and attitude of the informant. Limitations : 1 j u „ 1. This method can be used if the field of enquiry is small. It cannot be used when field of enquiry is wide. J'" SSf
m '30 Statistics for Economics-X 2.
It is costly method and consume more time.
3.
Personal bias can give wrong results
""erwise 5. This method is lengthy and complex ifSifMlSi m-smrnrn Merits : Kiai^ obtained from the third party, it is more or less free froJ 3
of the investigator and the inforr^ant I
3. It saves labour, time and money H iLtJ™"'"" """ '
i
of P'oHems can properl,'
Limitations : ar in to oh G( in( wt Mt Lin s-XI
Collection of Primary and Secondary Data
Thus, we find that both the above methods—direct and indirect personal interviews — have certain plus and minus points. For this reason the choice of the method depends on the nature of enquiry and sometimes we balance the demerits of one method by '.tsing the other method also for the same investigation. This way we can counter chec'. the data collected by one method with the other. (m) Telephone interview : The investigator asks questions over landhne telephone, mobile telephone and even through website. Various researchers, newspapers, television channels, mobile service providers, banks etc., use telephone service to get information from different people, e.g., exit poll, political or economical opinions, music or dance performance opinion etc. Even sometimes website or
internet are used for obtaining statistical data. These days online surveys through Short Message Service, i.e., SMS has become popular. Merits : 1.
Telephone interviews are cheaper than personal interviews.
2.
It can be conducted in a shorter period of time.
3.
The investigator can assist the respondent by clarifying the questions.
4. Sometimes respondents are reluctant to answer some questions in personal interviews. Telephone interviews are better in such cases. Limitations : 1. Information cannot be obtained from people who do not have their own telephones. 2. Reactions of respondents on certain issues cannot be judged; but it sometimes becomes helpful in obtaining information from respondents. (IV) Information firom correspondents : In this method, local agents or correspondents are appointed in different parts of the investigation area. These agents regularly supply the information to the central office or investigator. They collect the information according to their own judgements and own methods. Radio and newspaper agencies generally obtain information about strikes, thefts, accidents etc. by this method. It is adopted by Government departments to get estimates of agricultural crops and the. wholesale price index number. It is suitable when the information is to be obtained from a wide area and where a high degree of accuracy is not required. Merits : 1.
This method is comparatively cheap.
2.
It gives results easily and promptly.
3.
It can cover a wide area under investigation.
Limitations : 1.
In this method original data is not obtained.
2.
It gives approximate and rough results.
32 Statistics for Economics-XI f C
r fij 3. As the correspondent uses his own judgement, his personal b^as may affect the accuracy of the information sent. ^ nn^'^'TT
correspondents and agents may mcrease errors.
the mo
"
"
^^^ ^^^^ kept con^'entkllt '
Merits : ^^
^his method in cases where
informants are spread over a wide geographical area. o^p^eLri 1 ~
^^ ^^
-
-- Jess than the cost
3. We can obtain original data by this method Limitations : faift?"" ^^^ ^^^ informants. They may ques^r 3.
misinterpret or may not understand'some
There may be delays in getting replies to the questionnaires
4. ms method can be used only when the informants are educated or hterate so that ^ they return the questionnaires duly read, understood and answered 1 ' possibility of getting wrong results due to partial responses, and those IrmrnTre^uir^^ ^^ ^ ^^ ^^^ ^^e splcm: 6. There may be loss of questionnaires in mail. This method is suitable for the following situations • cTmpef LrS? '' "
questionnaire, Government agencies
compel bank and companies etc., to supply information regularlv to the Government in a prescribed form. ^ ^ regularly to the (b) This method can be successful when the informants are educated. inf( The be j org; and the high Mer Limit
1 3. 4. 5. 'Ki 33 Collection of Primary and Secondary Data Following are some suggestions for making this method more effective and successful. (a)
Questions should be simple and easy so that the informants may not find it a
m burden to answer them.
-ui
(b) Informants should not be required to spend for posting the questionnaires back therefore, prepaid postage stamp should be affixed. ic) This method should be used in a large sample or wide universe. (d) This method is preferred in such enquiries where it is compulsory by law to till the schedule. Thus, there is little risk of non-response. (e) The language of the schedule should be polite and should not hurt the sentiments of the informants. (VI) Ouestionnaire filled by enumerators : Mailed questionnaire method poses a tanber oi difficulties in collection of data. Generally, these filled questionnaires received to incomplete, inadequate and unrepresentative. S The second alternative approach is to send trained investigators^or enumeratoi^m M,rmants with standardised questionnaires wl.ich are to be fiUed^jn ^e im^estigator helps the informants in recording their answers. The invest^a^rs shoidd i honest tactful and painstaking. This is the most common method used by research iSons. They train investigators properly specifically for the purpose of an enqu^ ^d also tram them in dealing with different persons tactfu ly, to get Proper answers to Ac questions put to them. The statistical information collected under this method is
highly reliable. Merits : 1. It can cover a wide area. 2 The results are not affected by personal bias.
,-,.u
3' True and reliable answer to difficult questions can be obtained through ■ establishment of personal contact between the enumerator and the informant. 4. As the information is collected by trained and experienced enumerators, it is reasonably accurate and reUable.
,
5 This method can be adopted in those cases also where the informants are illiterate. 6'. Personal presence of investigator assured complete response and respondents can be persuaded to give the answers to the questionnaire. Limitations : ^
n■^
1. It is an expensive method as compared to other methods of primary collection of data, as the enumerators are required to be paid. 2. This method is time consuming since the enumerator is required to visit people spread out over a wide area. 3 This method needs the supervision of investigators and enumerators. 4" Enumerators need to be trained. Without good interview and proper traming, most ■ of the collected information is vague and may lead to wrong conclusions. 5. It needs a good battery of investigators to cover the wide area of universe and therefore it can be used by bigger organisations. I, If. P 34 Klot Suiwey or Pre-Te« • p
^f^^stics for Economics-Xi
a Pre-test or a guidihg survl;
" ^ -^uc.
mam survey. This is done to try out the auetlr
before starting th,
the general mformation about L po^Ja"^^^^ -thods for obtaL the pilot survey helps in : ^ ^e sampled. The information supplied b, [i) Estimating the eosr of ^ ■ avaihbiii,® of fc ""
rr.
'he ei™ needed for
or fan.
0-t.ons and a,so .„ .He ..proje.enr „ (w) Training of field staff "
rX"
--
^
namr,
casus MID SAIWnE SURVEVS ia) Census method/Census Survey, and (b) Sample method/Sample Survey .e^VS^td^^.—^ " —
Have a Cear n„dersra„d.n, „,
Population and Sample o all ,he ten,. A par, of .hVlorplZL ekcon termed as sampling. SupposeTe'^are 500
1 "T'" '
School. If we want to know the average wekta
Secondary
W'll get the mformation abont all the fLrhnnd f
each girl and
obtamed by dividing the total weigte of he tn' "
we.ght willbe
CX)VERNMENT SENIOR SECONDARY Ii f( P O £ S< UI gro' Sun
20 Source s-XI
Collection of Primary and Secondary Data
by taking only 50 girls out of 500 and obtain the average of this part of the total population. The average of 50 girls reasonably be representative of average weight of 500 girls. In this case weight of 50 girls is the sample. Census Surveys The objective of a census method or complete enumeration is to collect information for each and every unit of the population/universe. In this method every element of population is included in the investigation. Thus, when we make a complete enumeration of all items in population, it is known as 'Census Method" or 'Method of Complete Enumeration'. In above example, collecting weights of all the 500 girls in Senior Secondary School is census method of collection where no student is left over, as each student is a unit. Following are few examples of census : 1. The population census is carried put once in every ten years in India. Most recently population census in India was carried out in February, 2001 by house to house enquiry to cover all households in India. 2. Demographic data obtained by census method on death rates and birth rates, literacy, work force, life expectancy and composition of population etc. are published by Registrar General of India. 3. The data relating to estimation of the total area under principal crops in India are obtained by using village records maintained regularly by Patwari. Let us review the following census data in the following Table no. 2 regarding relative growth of Urban and Rural Population in India obtained from Reports and Economic Survey 2002-2003. TABLE2 Relative Growth of Urban and Rural Population in India r-................. Year i f r UrhaiP Popuiatinn {tn itorpi) Rural PopttUuioti (m rmrei) Total Ptipuldtion (m Lrine») As Perceraage of Total Popukttidn Urban popuhtion 1901 2.58
21.25 23.83 10.9
89.1
1911 2.59
22.62 25.21 10.3
89.7
1921 2.80
22.32 25.12 11.2
88.8
1931 3.35
24.54 27.89 12.1
87.9
1941 4.41
27.44 31.85 13.8
86.2
Rumi PvpuUition
1951 6.24
29.87 36.11 17.3
82.7
1961 7.89
36.02 43.91 18.0
82.0
1971 10.89 43.93 54.82 19.9
80.1
1981 16.22 52.11 68.33 23.7
76.3
1991 21.76 62.87 84.63 25.7
74:3 .
2001 28.50 74.2
72.2-
102.7 27.8
Source : Census Reports and Economic Survey 2002-2003. if:. iti I7 I (1 36 Statistics for Economics-XI 74
" 'rr
^^
Of India's population. In 2001
74 2 crore persons, out of about 102.7 crore total population lived in around 5 5 lakhs' i: ifo^lVt 2
r
urban a;ea. Th
-
-- or uian areas
T ' Population of around 24 crores lived in
urban areas The urban population formed about 11 per cent and rural population 89 per r^om I
urban population had gone up to around 28 per cen
n 2001 while still over 72 per cent people lived m rural areas. The above table show the relative growth of rural and urban population m India since 1901. The net addition to rural population between 1991-2001 was 1133 crore while urban population increased by 6.74 crore persons. The decadal growth at frru^d in1he rlr" T -- ^ mcrease of 2 1 per cent in the growth rate of urban population m the decade ending 2001 over the decade^nding
SAMPLE SURVEYS We may study a sample drawn from the large population and if that sample is adequate representative of the population, we should be able to arrive at val 7corcSn Method of collecting of data. In above example, collecting the weights of 50 girls out of 500 girls m Semor Secondary School is sample method of collectiol In this method ew students as sample considered for our study. metnod tew Following are a few common examples of samplin- • {a) We look at a handful of gram to evaluate the quality of wheat, rice or pulses, etc A
^^^ '^^^ ^P^" ^1-tric bulbs out of each lot"
[c) A drop of blood is tested for diseases like malaria or typhoid etc ^ ^ fudtrnfof^'tC ^^^ »
P-^-ion for final
(.) Th^^^elevision network provides election coverage by exit polls and prediction is nnnT"'' T ""V^^^'^^ical termmology population or universe does not mean the total numbe of people m an area; it means the total number of observations or terns fn r::att- -- ^ — — ^^lected from^ a ~ methods of sampling ^^^Broadly speaking, various methods of sampling can be grouped under mam (a) Random Sampling, and (b) Non-Random Sampling. Collection of Primary and Secondary Data 37 Let us discuss now the various samphng methods which are popularly used in practice. MiTHODS OF SAMPLING i Random Sampling (a) -1
Non-Random Sampling ib) Simple or Unrestricted Random Sampling Restricted Random Sampling (f) Stratified Sampling (//) Systematic Sampling or Quasi Random Sampling (f/f) Cluster Sampling or Multi-stage Sampling (a) (to) (c) Judgement Sampling Quota Sampling Convenience Sampling random sampling Random Sampling is one where the individual units (samples) are selected at random. It is called as probability sampling. Random sampling does not mean unsystematic selection of units. It means the chances of each item of the universe being included in the sample is equal. The term 'Random Sampling' here is not used to describe the data in the sample but it refers to the process used for selecting the sample. Following are the methods of random sampling. Simple or Unrestricted Random Sampling This method is also known as simple random sampling. In this method the selection of item is not determined by the investigator but the process used to select the terms of the sample decides the chances of selection. Each item of the universe has an equal chance of being included in the sample. It is free from discrimination and human judgement. Random sampling is the scientific procedure of obtaining a sample from the given population. It depends on the law of probability which decides the inclusion of items in a sample. To ensure randomness, mechanical devices are used. There are t^vo methods ot obtaining the simple random sample. They are : (a)
Lottery Method, and
(b)
Table of Random Numbers.
(a) Lottery Method : A random sample can generally be selected by this simple and popular method. All the items of the universe are numbered and these numbers are written on identical pieces of paper (slip). They are mixed in a bowl and then there starts the selection by draw one by one by shaking the bowl before every draw The numbers are picked out blind folded. All slips must be identical in size, shape and colour to avoid the
biased selection.
IMH' 38 Statistics for Economics-XI metal pieces on which nuZ^tT
Th! d "" "--^en or
device and each time one piece comesrotated by a mechanical of digits, for instance if the numbe'Ts ZZ This^i^ IS us. m drawi::^^-^^:;
u '^^^
""^ber
--
-ry large the above procedures if the disks, balls or slips L not XrouThnf' ^^ T^^^^^ " been a marked tendency to usetSroTrindT^^^^ > T T' " ^^^^ ^as samples. A table of random digkst simply a
the purpose of drawing such ,
by a random process. The follLing of Som ^gS tt^^^^^^^^^^^^ ~ (.) Tippet. Random Sampling Numbers. There are 10^00 numbe^t^anged 4 digits MG. Kendall and Babington Smith's Random Sampling Numbers, having 1 lakh ic) Rand Corporation's a million random digits (d) Snedecor's 10000 random numbers. ie) Fisher and Yates Table having 15000 digits Rc 2952 3170 7203 3408 0S60 Tippett Numbers 6641 3992 9792 5624 4167 9524 • 5356
1300 2693
2762 3563 1069 5246 1112 9025 ho
Th hoi ave strs ;san [peo 7969 5911 1545 1396 2370 7483 5913 7691 6608 8126 college. We will first nuX aVMOo Tui" f ''"t"'' ^ students, now we will cons J a pag^of ?fp
i'" ^^^ ""'"''"ing the
15 successive number either horLLhy or ^^a^ ""
^^
Merits '
w a universe. There are less
selected.
^
bas equal chance of being
Regularity begin to operate
^^^ ^aw of Statistical
( [Meri 1. 2. 3. 4. 5. 6. s-XI
Collection of Primary and Secondary Data
3. This method is economical as it saves time, money and labour in investigating a population. 4.
The theory of probability is applicable, if the sample is random.
5.
Sampling error can be measured.
Demerits 1. This requires complete list of population but up-to-date lists are not available in many enquiries. 2. If the size of the sample is small, then it will not be a representative of a population. 3. When the distribution between items is very large, this method cannot be used. 4. The numbering of units and the preparation of the slips is quite time consuming and not economical particularly if the population is large. t Restricted Random Sampling They are as follows : (t) Stratified random sampling : In this method the universe is divided into strata or homogeneous groups and an equal sample is drawn from each stratum or layer at random. This method is therefore useful when the population of the universe is not fully homogeneous. For example, suppose we want to know how much pocket money an average university student gets every month will be taken equal sample from various strata, namely : B.A. students, M.A. students and Ph.D. students etc. Stratified random sampling is widely used in market research and opinion polls, it is fairly easy to classify people into occupational, economic, social, religious and other strata. There are different types of stratified sampling {a) Proportional stratified sampling is one in which the items are taken from each stratum in the proportion of the units of the stratum to the total population. (b) Disproportionate stratified sampling is one in which units in equal numbers are taken from each stratum irrespective of its size. (c) Stratified weighted sampling is one where units are taken in equal number from each stratum, but weights are given to different strata on* the basis of their size. Merits 1. The sample taken under this method is more representative of the universe as it has been taken from different groups of universe. 2. It ensures greater accuracy as each group (stratum) is so formed that it consists of uniform or homogeneous items. 3.
It is easy to administer as universe is sub-divided.
4.
Greater geographical concentration reduces the time and expenses.
5.
For non-homogeneous population, it is more reliable.
6.
When original population is not normal (skewed), this method is appropriate.
Statistics for Economics-XI V\L N m. 40 Demerits 1. Stratified sampHng is not possible unless some mformation concerning ti population and its strata is available. concerning u 2. If proper stratification is not done the sample will have an effect of bias. If differ, strata of population overlap such a sample will not be a representative one («) Systematic sampling or quasi-random sampling : Systematic sampling is a simo by preparing this list m some random order, for example, alphabetical order SMnlr U
the list, « stands for any numl
Suppose we have a universe of 10,000 items and we want a sample of 1000, then ^^^ « - 10 The method of selecting the first item from the list is to decide at random f^^t ?hen th "r "'Tu'
Suppose we pTck up Z Z t
Then the other items will be 15th, 25th, 35th, and so on unSl we have got oVr fuH sal fullv rw'' T I u"
"
that the list of the univers!
fully random and that there are no inherent periodicities in the list. Merits 1. It ^yystematic, very simple, convenient and checking can also be done quickl] 2.
In this method time and work is reduced much.
3.
The results are also found to be generally satisfactory.
Demerits random will not be a determming factor in the selection of a sample. 2.
It IS feasible only if the units are systematically managed
3.
If the universe is arranged in wrong manner, the results will be misleading |
to divL and sub
"
^^
to divide and sub-divide a universe according to its characteristics. Thus if a survev ki be conducted in a country it will first be divided into zones or states l region t^^^^^ mailer units cities towns and villages and then into localities and hLseToW; At Jd non-ranoom sampun6 ccessi 3n No. Date.^............ s-XI the :rent tipler on is ieved mber. : take from item, mple. irse is ickly. !on as It m e have y is to ;n into .t each nethoc the hsl of the s, non-ing are Collection of Primary and Secondary Data (a)
Judgement or purposive sampHng
(b)
Quota samphng
(c)
Convenience sampHng
'n Judgement Sampling
\
This is also called purposive or deliberate sampling. In this method individual items of sampling are selected by the investigator consciously using his judgement. Therefore, it requires that the investigator should have a good knowledge of the universe and some experience in the field of investigation. Obviously, the choice of samples will vary from one investigator to another. For example, from a universe of 10,000 ladies who use a particular brand of hairdye, the investigator will select a sample of say, 1,000. His choice of this sample will be such that it is irrespective of the universe. For this an exercise oi judgement is required. In order for the judgement sampling to be reliable, it should be free from individual lies or prejudice. Since the choice of sample is not based on probability it does not guarantee accuracy and it makes detecting of sampling errors difficult. However, this methods is useful in solving a number of kinds of problems in universe and economics. The purposive or judgement sampling is suitable in the following conditions : (a) The number of items in the universe is small to which some items of important characteristics are likely to be left out. (b)
When small sized sample is to be drawn.
(c) When some known characteristics of the universe are to be intensively studied. (d)
It is also appropriate for pilot survey.
Quota Sampling It is a method of sampling that saves time and cost and is commonly used m surveys of political, religious and social opinion.- Interviewers are allotted definite quotas of the universe and they are required to interview a certain number from their quota. Quotas are decided on the basis of the proportion of persons in various categories. In other words, the investigator is given instructions about how many interviews should be taken say in a given localitv and what proportion should be from say upper, middle and lower mcome groups, as by some other classification which is predetermined. For example, for a study of truancy (running away) from school in Delhi the investigators are allotted quotas of say 10 schools each out of which two should be public schools (Boys), one public school (Girls), three Boys' Senior Secondary Schools, two Girls' Senior Secondary Schools, two Co-education Schools and from each school he is asked to interview 50 students, taking 10 students each from Classes VIII, IX, X, XI and XII. The interviewer can select any 10 students according to his own judgement. It is a kind of judgement samphng and provides satisfactory results only when interviewers are carefully trained and personal prejudice is kept out of the-process of selection. ' '
hi !i< 42 , Convenience Sampling
Statistics for Economics-XI;
P0.1:;: f'^intfri:- l^essr
~ -ce ^
example, for the study of truanrvTr a school or schools in I neiS^
^^e basis of convenience F.
^^^^^ ^^e invesdgat™selec
schools. This method is used wLn^e " .V ^^ ^^ convenient for hL to g^trthe not clear or complete source hst is t^a^lbl" T ^^e sample unit i easily available lists, such as teleDhLTW
may be obtained W
results obtained by' this m^aTntl^^^^^^ unsatisfactory.
^^^ ^ruly representatives of the universe and are
^^HIAGESOF (I •
of dae. bee. Jo,
getting quick results.
^
therefore, sampling is very useful in
sxhris fre^ir" —a, . for fc cV^ "^dr m - « method. 2
in some ways more reliable than cenLs
aliow a samplmg mefcd t
« fc o„,, pos.b,e or E ~ or bote, ma„ufaeLd"« fcTar:^;-,,*^ ^dl"' sampling method. '' P^^^'^le due to the scientific nature of » appropriate «e,d . neceiary^Srfctr:?"^^^^^^^^ le otl: Thi the larf Statistic The means a
Collection of Primary and Secondary Data 43 ^^i^lity of sample data ; The main purpose of sampling is to collect maximum information with minimum ^nditure of money, time and labour and yet achieve a high degree of -^curacy and Ability. For ensuring reliability certain principles must be followed. In samphng method : is presumed that whatever conclusions are drawn from a sample are also true for the lole population. This presumption is based mainly on the followmg two laws : (a)
The Law of Statistical Regularity, and
(b) The Law of Inertia of Large Numbers. .'r u (a) Law of statistical regularity : The law of statistical regularity is derived from the mathematical theory of probability. It says that a comparatively small group of items chosen at random from a very large group will, on the characteristics of the large group. Basically, it applied to rWom se^lection. Thus so in the process of sampling each unit of the universe has an equal chance of being selected. Therefore, the selected items can be said to be representative of the universe. Although the law is not as accurate as a scientific law is, it does insure a reasonable degree of accuracy. Since there is a certam regularity m natural phenomena, we assume a certain uniformity in nature A random samphng is said to follow the law of statistical regularity because of this basic uniformity m a universe.
r , -t- r
lb) Law of inertia of large numbers : This law is also called the law of stability of mass data. It is based on the law of statistical regularity. Basica ly, it states that if the numbers involved are very large, the change in a sample is likely to be very small in other words, the individual units of a universe very continually but the total universe changes slowly. That is, large aggregates are most stable than «tnaU Because of the slow change in the nature of total universe this law is called the law of inertia (laziness) of large numbers. For example, sugar production of factory will vary significantly from year to year but Ac sugar production of a country as a whole will remain comparatively s able. Or a g eat Inge may take place in the male-female ratio of family may appreciably -bange ove a short period, but the male-female ratio of a country as a whole will ^^
the period, ^o take another example, if a. coin is tossed 6 times we may get heaj^s f^r ^ Js and tails two times. But if a coin is tossed 5^0 times^ there is a high p^^i^ of getting heads and tails 2,500 times each. This happens due to ^^^ I oplation of this law. That is, when one part of large group is changing m one direction the other moves in the opposite direction. Thus, reliability of sampling depends mainly on randomness of selection of data and the large size of universe, expressed by the above two laws. Statistical Errors There is a great difference in the meaning of mistake and error in statistics. Mistake imeans a wronfcalculation or use of inappropriate method in the collection or analysis 44 Statistics for Economics-X. other words, the difference between the approximated (estimated value) and the actual value (true value) is called statistical error m a technical sense. For examl we make a' estimation that in a particular meeting, 1,000 persons are there. But we clnt persons It may be wrongly counted, as 1,030. There is a difference of 30 between the estimate value and counted va ue. This difference is called '...or' in statistics. But w^en weTak* aTS''^r VrThey arl knowi as mistake . For example, there is a meeting, we sent a person to count the audience Sources of Errors Following errors are likely to occur in collection of data : ur'l^! origin arise on accoum of inappropriate definitions of statistical unit scale, or defective questionnaire etc. For example, wrong scale to measun meLl -'I height to nearest of inch or approximatrTh differences may also occur due to differences in measuring tapes due tc manufacturing defect. In Physics or Chemistry such errors of mLsurementrwlI occur while taking readings on various instruments. nZZ
incomplete data, madequat.
crsdonna " sample, non-response of respondent, incomplete answers i questionnaire, misinterpretation of questions in questionnaire, careless oi unqualified investigators, etc. '-diciesb oi| dr^o I'f f^
-"hmetic calculatio
due to clerical errors, arithmetic slips etc. by omitting some figure consideri wrong value, making wrong totals etc. by respondent L investigator thrjlta^"''''^'"''^''''" ' statisticians for misinterpret! Types of Errors (a) Absolute and relative errors : Absolute error is the difference between the actua true value and estimated approximate value while relative error is the raTo o absoS error to the approximated value. absolut Absolute error = Actual value - Estimated value Symbolically, Ue = U' -U wr the enu mei wh( faul Relative error = Actual value - Estimated value Symbolically, Estimated value e= U'-U U Sec furthei obtain alreadj are inv b) unf s-XI
Collection of Primary and Secondary Data
Here, Ue = Absolute error e = Relative error U' = Actual value U = Approximate value niustration. Sales of commodity approximated Rs 497 and actual sale Rs 500. Absolute error (Ue) = 500 - 497 = 3 500-497 3
and Relative error (e) = .006
500 500 Relative error (e) can also be represented in percentage X 100 = 0.6%. 500 Relative error is generally used in statistical calculations because absolute error gives wrong or misleading calculations. (h) Biased and unbiased errors : Biased errors arise due to some prejudice or bias in the mind of investigator or the informant or any measurement instrument. Suppose the Hiumerator used the deliberate sampling method in place of simple random sampling method; then it is called biased error. These errors are cumulative in .ir re and increase when the sample size also increases. Biased errors arise due to fauli^ j^iocess of selection, faulty work during the collection of information and faulty method of analysis. Unbiased errors are not the result of any prejudice or bias. They are those which arise acccidently just on account of chance in the normal course of investigation. Unbiased errors are generally compensating. (c) Sampling and non-sampling errors : The errors arising on account of drawing inferences about the population on the basis of few observations (sampling) are called sampling errors. The errors mainly arising at the stages of ascertainment and processing 'of data, are called non-sampling errors. They are common both in census enumeration and sample surveys. To avoid these errors, the statistician must take proper precaution and care in using itfie correct measuring instrument. He must see that the enumerators are also not biased. Unbiased errors can be removed with proper planning of statistical investigations. Statisticians should have none of these errors. i" how secondary data is collected Secondary data are those which are collected by some other agency and are used for i^her studies. It is not necessary to conduct special surveys and investigations. We can obtain the required statistical information from other institutions, or reports which are ^eady published by them as a part of their routine work. It saves cost and time which 'are involved in collection of primary data. Secondary data may be either (a) published or (fc) unpublished. 46
Statistics for Economics-XI Ji hm. Sf. Published Soiu-ces The various sources of pubhshed data are as under : (/■) Gove^ent pubUcations : Different ministries and departments of Central ar State Governments publish regularly current information along with statistical da on a number of subjects. This information is quite reliable for related studies. ^ examp es of such publications are: Annual Survey of Industries, Labour Gaze Agriculture Statistics of India, Indian Trade Journal, etc. («) Publications of international organisations : We can obtain valuable internation s atistics from official publication of different international organisations, like, ti United Nations Organisation (UNO), International Labour Organisation (ILO International Monetary Fund (IMF), World Bank, etc. . (Hi) Semi-official publications : Local bodies such as Municipal Corporations, Distri Boards etc: publish periodical reports which give factual information about heal sanitation, births, deaths etc. (iv) Reports of committees and commissions : Various Committees and Commission are appointed by the Central and State Governments for some special study an recommendations. The reports of .uch committees and commissions contai valuable data^ Some of the reports are : Report of National Agricultu Commission, Report of the Tariff Commission, the Patel Committee Report e (v)
Private publications :
(a) Journal and new^apers. Journals like Eastern Economists, Journal of Industr and Trade, Monthly Statistics of Trade; and newspapers, like Financial Expres Economic Times, collect and regularly puWish the data on different fields ( economics, commerce and trade. (b) R^earch institutions. There are a number of institutions doing research o allied subjects This is the most importarn source of obtaining secondary dat The National Council of Applied Economic Research and Foundation ( !>cientihc and ^onomic Research are such institutions. Research scholars at ti university level also contribute significandy to the availabihties of secondai (c) Professional trade bodies. Chambers of Commerce and Trade Associatio, publish statistics relating to trade and commerce. Federation of Indian Chamb of Commerce, Institute of Chartered Accountants, Sugar Mills Associatio Bombay Mill
Owners Association, Stock Exchanges, Bank and Cooperath Societies, Trade Unions, etc. pubhsh statistical data. (d) Annual reports of joint stock companies are also useful for obtaining statistic information. These are pubKshed by companies every year. ^^^ ^md^'' s-XI
also, provide valuable data for reseat
Collection of Primary and Secondary Data
Unpublished Data Research institutions, trade associations, universities, labour bureaus, research workers and scholars do collect data but they normally do not pubHsh it. Apart from the above sources we can get the information from records and files of government and private offices. Limitations of Secondary Data One should use the secondary data with care and full precaution and should not accept them at their face value as they may be suffering from the following limitations: 1.
They may not have been collected by proper procedure.
2. They may not be suitable for a required purpose. The information which was collected on a particular base may not be suitable and relevant to an enquiry. 3. They may have been influenced by the biased investigation or personal prejudices. 4.
They may be out of date and not suitable to the present period.
5.
They may not satisfy a reasonable standard of accuracy.
6.
They may not cover the full period of investigation.
Precautions in the Use of Secondary Data The investigator should consider the following points before using th j secondary data : (a) Are the data reliable? {b) Are the data suitable for the purpose of investigation? (c)
Are the data adequate?
(d)
Are the data collected by proper method?
(e)
From which source were the data collected? if) Who has collected the data?
(g) Are the data biased? . Thus, the secondary data should not be used at its face value. It is risky to use such statistics collected by others unless they have been properly scrutinised and found reliable, suitable and adequate. ■ ■ ijl^ofrlant sources of secondary dali|> of india and national survey organisations) There are various sources and organisations through which statistical data are being compiled in India. Since India achieved Iiidependence, great and rapid strides have been made in the field of collection of data. In the context of economic planning, importance of statistics (data) in the country has become great. Statistics are necessary for framing and judging the progress of economic planning. The study of Indian statistics is made under following heads : I.
Statistical Organisation of India (CSO)
II.
Indian Statistical Material.
48 Statistics for Economics-XI This can be studied >finder following sections : (A) Agriculture-Statistics (B) National Income and Social Accounting (C) Population Statistics
(D) National Sample Survey
(E) Price Statistics (F) Industrial Statistics (G) Trade Statistics (H) Financial Statistics (I) Labour Statistics There are some agencies both at the national and state level, which collect, process .^nd tabulate statisticar data. Some important major agencies at the national level are ^ensus of ^dia, Narionai Sample Survey Organisation (NSSO), Labour Bureau, Central Statistical Organisation (CSO), Registrar General of India (RGI), Director General of Commercial Intelligence and Statistics (DGCIS), etc. census of india unique experience of undertaking the biggest census in the world in 1981 and has also an unbroken record of more than hundred years of decadal censuses Ihe Indian census is universally acknowledged as most authentic and comprehensive source of information about our land and people. In 1869 Hunter was appointed Director General of Statistical Surveys. He not only elaborated the statistical system but also assisted the statistical surveys of districts and provinces. That later followed into
famous Gazetteers. He advised m conducting of census of India which undertook explanatory surveys from 1869 to 1872 and thereafter matured into a decennial census which ever since contmued without interruption. After 1872 the next census was taken in 1881 and ^nce then it has ^become a regular feature of holding census every ten years uninterruptedR The Census of India provides the most complete and continuous demographic record of T
Independecne was held in 1951 and latest one completed
m .001. The study of population is important for several reasons in overall study of economic development. Information of demographic characteristics include birth and death, fertility, sex ratio, age-composition, migration and literacy etc. The economic Characteristics of ppulation are manifested through workers' participation m economic classification of workers m various occupations, employment The data generated by the Census of India 2001 provide benchmark statistics on the people of India at the beginning of the next millennium. This is a mirror of a fair ^presentation of the socio-economic and demographic condition of our people which constitute about one-sixth of the human population on this planet. The census statistics s useful for assessing the^impact of the developmental programmes and identify new thrust areasTor focussing the efforts on improving the quality of life in our country Basic population data fmm Primary Census Abstract. Census of India 2001 gives information ot population m India as : TABLE 3 Persons
Males Females
Sex Ratio
1,028,610,328 —^- ^ _" 532,156,772 496,453,536 933 Collection of Primary and Secondary Data 49 national sample survey organisation (ii5s0)
The National Sample Survey (NSS), initiated in the year 1950, is a nationwide, large scale continuous survey operation conducted in the form of successive rounds. It was established on the basis of a proposal from Prof. P.C. Mahalanobis to fill up data gap for socio-economic planning and policy making through sample surveys. On march 1970, the NSS was recognised and all aspects of its work were brought under a single Government organisation namely the National Sample Survey Organisation (NSSO) under the overall direction of a Governing Council to impart objectivity and autonomy in the matter of collection, processing and publication of the NSS data. The Governing Council consists of 18 experts from within and outside Government and is headed by an eminent economist/statistician and the member-secretary of the council is Director General and Chief Executive Officer of NSSO. The Governing Council is empowered to take all technical decisions in respect of survey work, from planning of survey to release of survey results. The NSSO headed by a Director General and Chief Executive Officer, has four divisions namely. Survey Design and Research Division (SDRD), Field Operation Division (FOD), Data Processing Division (DPD) and Coordination Publication Division (CPD). A Deputy Director General heads each division except FOD. An Additional Director General heads FOD. Functions of NSSO The functions of National Sample Survey Organisation are : (i) Collection of data on socio-economic conditions, production of small scale household enterprises consumption etc. on continuous basis in a comprehensive manner for whole country. A major objective of NSS has been to provide data required to fill up the gaps in information needed for estimation of national income. [ii) Collection of data relating to the organised industrial sector of the country. {Hi) Supervision of surveys conducted by states in agricultural sector through their own agencies and also giving guidance to them for analysing and coordinating the results of these surveys. The NSSO took a forward view of the data requirements to planners, research workers and other users and draw up a long term programme. The programme conducts periodical surveys on : {a) Demography, health and family planning; {b) Assets, debt and investment; (c) Land holdings and livestock enterprises; {d) Employment and unemployment, rural labour and consumer expenditure; and (e) Self employment in non-agricultural eflterprises. The data collected by NSSO surveys on different socio-economic subjects are released tiirough reports and its quarterly journal 'Sarvekshana\ The data comprises different iocio-economic subjects like employment, unemployment literacy, maternity child care.
■1tr 50 Statistics for Economics-XI utUisat^n of public distribution system, utilisation of educational of services etc Th car Aoar; fTolfll . T 2004)-was on morbidity and head care. Apart from collection of rural and urban retail prices for compilation of consume pn.e mdex numbers NSSO also undertakes field work of Annual S^ ^dust^ conducts crop estimation surveys. ^ maustries an exercises 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. i-XT What do you understand by Statistical Enquiry? Explain d~ft~f
^
merits anc
Discuss the comparatwe merits of various methods of collecting primary data. SrsTable^^^^^^^^^^^ "
'
i-tigations-,
What are the similarities and dissimilarities between the two methods-l questionnaires to be filled in by informants and schedules to be fild in h enumerators? Explain with examples. * mat is a questionnaire? Give a specimen of a questionnaire. Describe the questionnaire method of collecting primary data. What precaution! must be taken while preparing questionnaire? precautionf Write short notes on : (a) Census of India {b) National Sample Survey Organisation (NSSO) mat IS Secondary^data? Discuss the various sources of collecting secondary data, mat precaution should be taken before using secondary data? Explain iTZn 1 constructing interview schedules and questionnaires F ame at least four appropriate multiple choice options for following questions (t) How often do you use computers? («) What is the monthly income of your family? (m) Rise in petrol price is justified : (iv) Which of the newspaper do you read regularly ? Jv) Which of the following most important when you buy a new dress' ls and -years' or 'months "n X-ax,. 4 Scak ■ A diagram should be drawn with the help of geometric - ^ scale slould be selected to su.t far as possible be in even numbers or multiple ot 5, lU, /u, zo, luo 'months' on X-axis. . ^u v onH Y . A i^A^^ ■ The 'sc/j/e' of measurement on both X-axis ana i sl'zrrrf ^^ « f U t^aror^rVftX™ ^nJt through different colours, shades, dotting, crossing, etc., an index must g for identifying and understanding the diagram. the source from which data have been obtained, more effective than a complex one. types of diagrams There are various types of geometric forms of diagrams used in practice as shown on following two Geometric forms of diagrams : A.
One-dimensional Diagrams
B.
Pie Diagram
A ONF-DIMENSIONAL DIAGRAMS ™lionaMiagrams are also called ^^^^ JthXiror:! used in practice. They are called one-dimensional because of height of the bar
90 significance and not the width of the bar Foil " u (-) Simple bar diagram (b)
^^
j
^^^ ^ypes of bar d^
Sub-divided bar diagram
(c) Multiple bar diagram id) Percentage bar diagram (e) Broken bar diagram (/) Deviation bar diagram (a) Simple Bar Diagrams • The variable can be presented, A Jimple^rdlTlrc^ b'^ vertical base. It is used for vistil ?
T
"" horizontal o,
production, population, sa,es,Xt e^dilr one category either in years, months w«ks et T or groups. All the bars L be brau^LZ attractive.
^fonnation of
Tu
or shading to make them more
simpfc bar diagram the scale is detet^ned'^teiTof ^ Illustration 1. Draw a b«r A:,
""
of computer softw^r
relating to expo,;
-Xc^ore. ; T/oo'
"
" the series,
--- 2000.. ' .OOf-0.
, &o„omfc Survey. 2002-03 p 144) Solution. ' '
export of computer software (1997-2002) Scale : 1 cm = Rs 7,000 crore Y 42000 35000 ff 28000-S O r 21000^ 140006,500
36,500 28,350 17,150 10,940 YEARS Fig. 1 2001-02 "tr rrr aw——— ; vertical base showing horizontal b«s as under : Alternative solution - Vertical base.
^ „ V ^vU
w T Years on X-axis; Value (Rupees m -^ores on Y-ax s W 2 • Years on Y-axis; Value (Rupees m crores) on X-axis. Scale : 1 cm = Rs 7,000 crores. Export of Computer Software (1997-2002) Scale : 1 cm = Rs 7,000 crores 2001-02 2000-01 1999-00 1998-99 1997-98 0 36,500 ■ 28,350 4 17,150 10,940 . 6,500 —r 7000
14000 21000 2^00 3M00 42000 Rupees (in crores) .. above ™o „e ui^L ^ - ^^^^ i„ the export of computer ^^^^^^^ the software export have ,crores) m 1997-98 to Rs per year for last four years. -TwoXtl^rr;!: o"^^^^ .o^ -nomlc survey .00.0. are ,,
....______Dri>>a nhanaes
given below :
Poodgrains Production
(in million tons) Wholesale Price Changes 52-WeeteAverageJnflat^^ '1999-00 -01 -02 -03 -04 X (Provisional) Average up to Jan. 14, 2006. e First advance estimates (Khanf only). .-05 -06 1999-00 -01 -02 .03 -04 -05 -06 Fig. 3
92 ReauirPnt^^t j
'
.he guZrr/a Sir
given
^^^^t^stics for Economics-X
(h) Sub-divided Bar Diaeram • TJ, j In general sub-divfdedo;!^^^^^^^
^Component Ba
values of the given data is to be dividedTo v^no F-t of all a bar representing total s CwrXn
'^e to.
Z ^^
proportion to the values given in the dat^ S/ ? be'used to d^sttguish-os-g
, P^^s i, dotting or designs can
remember that the various componentrshouFd be t '
tndex' IS to be given alongwith the Lgram to ? Illustration 2. Draw a snir.Kl ^
" ^^^^ bar.
differences.
Draw a suitable diagram to represent the following mfo^ation : Year 2001 2002 2003 2004 2005 Trains Murder 108 131 97 102 75 Solution. Robbery 82 115 144 70 68 Loot 321 386 352 285 245 Total 511 632 593 457 388 CRIME IN RUNNING PASSENGER TRAINS (2001-2005) Scale : 1 cm = 200 crimes 800 600co m E - 400-1 O a loot e robbery ■ murder
200 - _
' ^^ia
2001 pnno -----^-2002 2003 YEARS 2004 jr 2005 Fig. 4
(c)M between t inter-relaC of drawin; In this cai spacing isi in a set, d be given. ' 93 Qigrammatic Presentation ^ifU^w^ofD^riation S.E. Asia West Asia Africa Other Regions Total d,agram to represent the above data. Ltion. Suh-d.v.aea bar a,a.ta,n .s sn,.ab.e to the ahove data. YA 100 Q Other Regions ® Africa pfi West Asia o-E. Asia 2003-04 2004-05 YEARS Fig. 5 94 ,
Statistics for Economics-}
Illustration 4. Draw a suitable diagram of the following data :
Statement of CrimeJnR^g Passenger Trains Solution. Year
Murder
Robbery
2001 108
82
321
2002 131
115
386
2003 97
144
352
2004 102
70
285
2005 75
68
245
Loot
CRIME IN RUNNING PASSENGER TRAINS (1998-2002) (Scale ; 1 cm = 100) 500 4001 2 300H tr o 200 looses 321 352 285 B Murder ■ Robbery Q Loot 245 tons during the same fortnight last vear(ronnf TK « ? T' during the first fortnight of DecemberToo ^ 2 ssloo^f
f""
and 41,000 tons for exports as against 1 54 000 ton /
.
consumption
exports during the sam"! fortnigriast sfasfm^ (t) Present the data in a tabular form
(Hi) Present these data diagrammatically. 95 Digrammatic Presentation Solution. (/) Presentation of data in a tabular form.
•c
i Stock
Fortnight Sugar Production, Off-take for Internal Consumption, Export arU Stock in Sugar Mills in India. December, 2000
(figure in thousand tons) December, 2001 , (First fortnight)
Production Off-take from Mills Export Stock Source : muiau du^^i
378 154 Nil 224
387 283 41 63
---------
export and stock which we have calculated. (Hi) Diagrammatic presentation of above data by (a)
Sub-divided bar diagram
(b)
Multiple Bar diagram
INDIAN SUGAR MILLS ASSOCIATION REPORT (Fortnight Sugar production, off-take for internal consumption, export and stools in Sugar Mills in India Scale : 1 cm = 50,000 tons.) MULTIPLE BAR DIAGRAM SUB-D'VIDED BAR DIAGRAM 400350300250iction K 8,000 1
200-
tories ■
150-
iption 1 lil for ■
100-
50grams B
0-
Dec. 2000 Dec. 2001 (First fortnigh) (First fortnight) Dec. 2000 (First fortnigh) Dec. 2001 (First fortnight) Fig. 13 96 Statistics for Economics-XI j Proceeds per Chair Factory A (Rs)
Factory B (Rs)
Wages Material Other Expenses 160 120 80 200 300 150 Total Selling Price 360 400 Profit or Loss (±)
650 600
(+) 40(-) 50
1 he percentages are calculated as under : Ptrcentaj^^ (For Percentage Bar Diagram) Proceeds per Chair i-______
Factory A (%)
j Wages 1 Material Other Expenses Total Selling Price 90 100 Profit or I oss (+)
(+) 10
OT UJ 4 m ^ a. 3 tr 97 Digrammatic Presentation % COST Y
40 30 20
108.3 100
Factory B (%) 33.3 50 25
chmr 3. pnofit amo loss 100^ , , . n » PROFIT AMD LOSS : 1 cm = 20% factory b factory a (t UJ
60-
QZ CO
40"
UJ lU Qrj
20'
cc o: -20 M other Expenses Q Material Wages wm Profit or Loss Fig. 8 . c .pries in which some values may Broken Bar Diagram : Sometimes we may S™^ reasonable shape each bar is written on the ™ ° , b„ a suitable dragram. Year
Number of students
2001 2002 2003 2004 ^___-
25 48 375 125
neces^to brsn^ -^ ^ 98 Statistics for Economics-XI 3 NO. OF STUDENTS .N SCIENCE (2001-2005)
Scale : 1 cm = 25 students 200175f2
150-
2 LLI Q 125? CO
100-
u. o d
75-
2 50250-2002 2003 2004 YEARS 2005 Fig. 9 net 'i^r: ^e^l^ipt tt^ t.e export, etc., wh,ch have both
" "" "
■n plus and minus values to plot th.ron the base l.ne and negative vales bell'fbatTe" Year 1998 1999
2000 2001 2002 Export 47 125 20 94 120 Import 30 115 39 no 125 (Rs in Lacs) Balance of Trade
17 10
/ -19 -16 -5 99 Digrammatic Presentation . Solution. BALANCE OF TRADE (1998-2002) Scale 1 1 cm = 5 lacs Y 25 -20 -15 -I CO O CO lU LU ti. =3 CC 10 5 0 -5 -10 -15 -20 -25 gg Surplus ■ Deficit
u t 1998 1999 2000 YEARS Fig. 10 2001 2002 B. PIE DIAGRAMS
_^
^^^^^ ^^ ^^^^^^^ These
Pie diagram or circular diagram is ^^^^
^ comparatively easier to draw,
diagrams are very useful in emphasising
exhibited. Circles can
With circles and sectors, totals as well as comp^em parts ca ^^^^^ be drawn by making the^/™
^^^^ ^^^
^IgllmL ^called I p.
angle at the centre is 360 or 2%. I hereio , _ ^ Pie diagrams are very POP^^-^J/JJ"
oercentage breakdowns by
represent the
portioning a circle into various parts ^ various parts will indicate Lvernment expenditure Transport,
and
different
pornons^^^^^^ ^^^^^^^^
the expenditure over different heads l^ke^ heads Namely, food, clothing. Education, etc. Similar^ expenditure of ^^^ ^^ components or the rent, education, etc. If the series is
diagrams are less effective than
difference among the components is very small, then pie a g bar diagram. Steps for Construction of Pie Diagram percentage of respective totals. Pie helpful for comparison. 100
- «ke„ ,o be equal to 360-. ,t ,s „™ te^e^ h" "" ry to express each part proportionately ™^rees.S.„ceiperce„totthetotaWalne,se,„alto^.3..,.hepercentages oahe .mponent parrs w„l be now converted to degrees by .n,t,ply.„g each of Degree of any Component part = Component value simultaneously for comparison tbrralTrh
"
proportional ro the square roots » ^ ■
^^^^^
-"-a.. . is common
position on the circle. Now, with this K ^^ o'clock centre with the help ofp^r t] "^a o t iTete to f ^
^ ^^^ component, the new line drawn a
circumference. The sector so obtained wi^
the
component. From this second line a ble nowT""'
^^^^^^^ ^^^
equal to the degree represented by second c^Z
^^ ^^^ ^--^re
the portion of the second component S^ih 7 representing component parts can be coZuTd '
-P-senting differen?
Be distinguished Illustration 9. Constmrr o a-break-up of the cost of — Item Expenditure Labour Bricks Cement Steel Timber Supervision % Ltgrammauc Fresentation _
^^^^ percentage into
So.».>o„. Before draw™, » 3.6" —, we Labour Bricks Cement Steel
25 % 15 % 20 % 15 % 10 % 15
Timber Supervision
Fig. 11 n-
of
three textile items in
percentage
Items Readymade Garments Cotton Textiles Wollens Textiles Years 2003-04 Total 52.2 19.1 28.7 2004-0S 100.0 / 41.7 23.3 35.0 100.0 102 Solution. (Degrees of angle are rounded off) Statistics for Economics-XI j Items Redymade Garments Cotton Textile WoIIen Textile Total 2003-04 % 52.2 19.1 28.7 100.0
Degree of angle 188 69 103 360 2004-0S % 41.7 23.3 35.0 100.0 Degree of angle 150 84 126 360 export of textile items 2003-04 2004-05 Fig. 12 Illustration 11. Represent the following data by a pte diagram. basfs of 360 r
,
basis of 360 taken as equal to the total of the values. Family X
Family Y
1. Food 2. Clothing 3. Rent 4. Education 5. Miscellaneous (Including Saving) 400 250 15r 40 160 640 480 320 100 60 Total 1000 1600 103 the digrammatic Presentation >ms of Expenditure Rs 1.
Food
2.
Clothing
3.
Rent
4.
Education
5.
Miscellaneous (Including Saving)
Total Square root 400 250 150 40 160 1000 31.6 400 —x360 = 144 Family Y Rs _ , ___i 1000 ^x360 = 90 1000 ^360 = 54 x360 = 14-4 1000 1000 1000 x360 = 57.6 360 640 480 320 100 60 1600 40 640
1600 480 x360 = 144 1600 1600 1600 1600 x360 = 108 x360 = 72 x360 = 22.50 x360 = 13.50 360 Radii of circle are determined m proportion 3.2 : 4 (31.6 : 40). Wore the radU of arcle accordmg to avaUabUtty of space 3.2 are : Family X : Radius -y = 1-6 cm 4 Family Y : Radius - = 2 cm expenorrure of family x and y Food im] Clothing m Rent B Education B Miscellaneous FAMILY X FAMILY Y Fig. 24 104 limitations of diagrammatic presentation Statistics for Economics-xl fo/lowng points „„„ remembered W—HmS-Ot^^^^^
" a .i.e,r capacity to g,ve
.n the,r basic fnncti^rrsytdTrj 2. Diagrams can show
mterpretation of diagrams, tlie
*
presentation.
and ™ tje dte^rb^Lrl^^^-Tt'''""'"-^' in diagrams.
facts are not possible to show
by tables etc. they can misrepresent facts diagram for visual Presentation of" n"^^
^^ ^ P-icular
and the object of presentation. Therefore, it shLld be m.^ t ^^^ data A well constructed simple and attractive Ltam sho
^are and caution,
easier to understand at a glance; sucrpresentatiotr^^ newspapers, magazines and journals
^^^ mformadon is
be seen in financial reports in
2. 3. 4. 5. 6. Questions : •
a-::
—
feplam the various rules of drawing a diagram. TpLin JT "ftheir utility, txplam M bar diagram, and (b) pie diagram Digrams are less accurate but more effective than tables in presenting the data • ' rc^mtlstc:: " —« "-wmg. W Composition of the population of Delhi by reltgion H Agnculture production of five states of India. What are the merits and limitations of dtagrammatic representation of stat,st.cal Explain the following with illustration ■ M Sub-divided bar diagrams, and W Multiple bar diagrams. 8 9. 105 iigrammatic Presentation
(Write short notes on the following 1(a) Percentage bar diagram , (c) Deviation bar diagram 1 (e) Multiple bar diagram. (b) Broken bar diagram (d) Sub-divided bar diagram iLt the following data by simple bar diagram. PRODUCTION OF COAL (Million Tons) Production pillion Tons) bar diagrams DEMAND AND AVAILABILITY OF STEEL (Thousand Tons) Exports (Rs in crores) 4. Present the following data by sub-divided bar diagram. total import
Food Fertilizers Mineral oil Others Total 2001-02 2002-03 474
795
125
298
341
1,113
2003-04
1,789 1,951 2,729 4,167 (Rupees in crores) 2004-05 988 323 1,042 2,394 4,744 / Represent the foJlowine • l . «e diagram. 'i-e He,p
for Economics-x\
Sunp.e Bar diagram. J
Yi^f
■ ■■ ■ .. ■ . ■:--—---
' 7. Export (Rs in crores) Import (Rs in crores) 2002 ~~2m 73
80
85
70
72
74
_2004
ZOOS
8. Food Clothing Rent Education Miscellaneous Farntly A (Rs) Family B (Rs) P'Xpettdiiure 9 TU --——i—^ 1440 ' ^'iree year's result of XTT ri T _ •
^-
".e fCowmg rab...
107 Ipigrammatic Presentation ntmaiic 11 B (Rs) 3
1
75
100
175
150
30
25
20
25
-----
Price per Unit Quantity Sold Value of Raw Material Other Expenses ________ Show the following data by percentage bar diagram^^
L^^
of a product . bar
chart : COST P^OTRPDS AND fROm AND LCTJ^ Cost per table : (a)
Wages
(b)
Other Costs
(c)
Polishing
Total Cost Proceeds per table Profit (+) Loss (-) Chapter 7 cmpbic presentation 3. sw Construction of Graphs Graphs of Frequency Distribution Line Frequency Graph • Histogram Frequency Polygon Frequency Curve : asiiiiiifa*
Graphic presentation gives i rr
-
as a tool of analysis. "
(Line Graphs,.
^^
... ------- of graphs;
109 Graphic Presentation
.„j
axes. (See Fig. D-
equal parts called qaadrants
.^j
of origin 'O' which represent^
Quadrants : G-ph pa^ys ^mded " ^^^
^^ ^ Y are posttrve.
Fig. 1 .ea. - - - ^^^ -V-Tdata. d.«e„„t .e^lu-'rS^^Str;: S^rrXs . ol t„ue, re—. ''Tt^ol senes graph X-axts ^^ ^
^^^^^^ ^
110 ■JJ
Statistics for Economics-XI
OF FREQOEIICr scale wid, d,e difference of lO^wS™ T ' " "" wasting too much of space of ^a7h pSr «« ^ S'^Ph require a lot of space so that X^is is
T",T'" "" ""
portion of the scL may be t use of 'kinke, fe' in ^phtc pr^tT^e Figi'r"" " ' frequency graph Scale : 0.75 cm = 10 Rs on X-axis 0.75 cm = 10 Workers on V-axis S'graphs..
axis
(b)
Histogram
(c)
Frequency Polygon
(d) Frequency Curve or Smoothed Frequency Curve ie) Cumulative Frequency Curve or 'Ogive' (a) Line Frequency Graph fluency array, on graph by which the line is drawn. represents the frequency of that variable on kaphic Fresentation 111 Heieht in incht
»
60"
90
61"
80
62"
120
63"
140
64"
132
65"
70
66"
40
Nc
-~d----f)
^ Metlwd {. X-axis for variables under study (Heights in inches) 2.
Y-axis for frequencies (No. of students)
3.
Draw a vertical line on each value equal to the length of each frequency
4. Both the axes must be clearly lebelled and scale of measurement clearly shown. X-axis can conveniently be determined according to the need of the problem. We can have three varieties of X-axis. Taking the above illustration they are : (a)
Use of kinked hne
(b)
Starting from 59"
,.^,.•
(c) Starting from 60" (use thick line to read the data properly). See the graphs given {d) Both axes must be clearly labelled and the scale of measurement should be clearly shown. Solution.
HEIGHTS OF STUDENTS
Scale
140 120 -
^
100 -
lU Q 3 is
80-
LL o
60 -
d
40 -
H 20 (a) using kinked line 1 cm = Frequency 20 Students 1 cm = r on X-axiis Fig. 13 il2 Statistics for Econo >mtcs~}
STARTING FROM 59" (X-axis) - iF-pu 20 StudL onV-axis ^ I cm _ 1 on X-axis Scale 1 starting FROM 60"
!
■ ^ = Frequency 20 Students on V^axls^ 1 cm = 1" on X-axis yf (b) Histogram '
which each
and also called a frequency histo^m : '' " ' ^^-dimensional diagram Cases of Constructing Histogram U) Histogram of Equal Class Intervals {« Histogram when Mid-points are given Histogram of Unequal Class intervals Method 1. X-axis for variables under study (Marks) .y-axis for frequencies are freq Thus is pn («)K n obtai 113 \ Graphic Presentation
class with frequency.
3.
oe. reW.
4.
Both the axes must oe ucdny clearly shown.
Solution.
—
histogram
Scale : 1 cm = 10 Marks on X-^is 1 cm = frequency 4 on Y-axis 1-
in for all the classes and the frequencies
In the above illustration 2, class mterval is 10 for all SSe!rr:
-Se for each class can be deaded(c^^
frequency) Class (Marks) 0-10 10-20 20-30 30-40 40-50 50-60 Class frequency (f) 4 10 16 22 18 2 10 X 4 = 40 10 xlO = 100 10 xl6 = 160 10 x22 = 220 10 xl8 = 180 10 Total Area = 720 Total frequency^^;^^______^ - ^ interval Histogram = ^en ^ff^^^Tora the following aisnr,bu,ion of total marks ojrr^-"'a'^Sjra Boara H—. 114 Method Marks (Mid-points) No. of Students (f) 150 160 170 180 190 200•8 10 25 12 7 3 Statistics for Economics-XI Graf mid-poirns
different classes from the given
2. X-axis for variables under study 5:
(Marks). 4 iTZ r fr^q^es (No. of Students).
with frequency
he clearly shown. ..... '
the measurement should
bet^ sS::^^ first mid-pomt, get the difference ^vide the difference by 2, /.e., 10/2 of
we get lower and upper lim.
=5~
Thus, the class decided is 145 ~to 155 ^ ^ = ^^^ - "PP- H.^" U^g the same -jet ^h^
mid-points as under :
No. of Students : g IQ
^25
^
^hri ways : ^
See the Figs, below : ^
^^^ Starting from 135 marks.
(iii) I ni Solution. histogram kinked line method Scale : 1 cm = 10 Marks on X-axis 1 cm = 5 Students on V-axis histogram x-axis-starting from 145 marks Scale : 1 cm = 10 Marks on X-axis 1 cm = 5 Students on V-axis 165 175 185 marks Fig. 7 Sd Nc histogi Metho 1. 2. 3. 4. 115 ^Graphic Presentation histogram X-AXIS^starting from 135 marks Scale 1 cm = 10 Marks on X-axis 1 cm = 5 Students on V-axis i UJ
o 3 u. O dZ 135 145 155 165 175 185 195 MARKS Fig. 9 205 215 S. Hit) Histogram of Unequal Class Intervals 'Tst^lo 4. Rep«se„. .he fonowing^^ans of of Workers .he Cass — are unequal, frequencies n.us. he adiusred, otherwise .he his.ogram would give a misleading picmie. '^"I'^Take .he class which has d>e lowes. class in.erval. 2. Do no. adius. .he frequencies of
i„,erval.
^rr^^^^^^ each .ecan^e of h.s.ogram hu. ■ widths will be according to class limits. 116 Thus, the adjusted frequencies are : histogram Scale : 0.5 cm = Rs 5 on X-axis 1 cm = 5 Workers on V-axis daily wages in rs Fig. 10 japhic Presentation Histogram : When Class Intervals are given by " Method ^ IllustrLn 5. Constru^^ ' r -Jji^s ^ Students W
117 5-9 10-14 15-19 20-24 25-29 30-34 4 17 25 32 13 6 Solution.
„ inrliisive method (where lower and upper
Note : Since the class intervals are given ^
.^d upper limits of
Adjustment : Find the difference between lower hmn^ _ and so on. Adjusted Class Limits Marks Students (f) 4.5- 9.5
5
9.5-14.5
17
14.5-19.5
25
19.5-24.5
32
24.5-29.5
13
29.5-34.5
6
histogram Scale : 1 cm = 5 Marks on X-axis 1 cm = 10 Students on V-axis 9.5 14.5 19.5 24.5 29,5 marks Fig. 24 118 (c) Frequency Polygon («) Without histogram. Statistics for Economic^y, Method
J ^^^^^^^^
loISo" --20-30 5 30-40 12 40-50 15 50-60 22 60-70 14 4 —" - !>uitaDJe histogram kepnm„ 2- the of the '"Ho " ™T
""""P'-
3. Jotn these md-p„i„„ „f ^.de of each rectangle ciearly sro^^^"^' ^^^^^ ^^^elied and the scale of the meas Solution. measurement should histogram and prequencv polygon JO on X-axis 1 cm = 5 Students on y-a*is 25' CO
20-
& UJ s
15-
1co u. o
10-
d 2 50- Histogram
-Frequency Polygon Fig. 24 Graphic Presentation
jj^
While drawing the frequency polygon, we observe that some area which was under the histogram has been excluded and some area which was not under histogram has been included under frequency polygon. This dotted area which was under histogram but is not under the frequency polygon. This dotted are is excluded from the area of frequency polygon. But the shaded area has been included under the polygon. This was not under histogram. Thus there is always some area included under the frequency polygon instead ot the area excluded from histogram. Therefore, the total area excluded from the histogram ts equal to the area mcluded under frequency polygon. (ii) Frequency Polygon : Without Histogram und™"^ ^^^ illustration, we can get the frequency polygon without histogram as Method 1.
Take the mid-point of each class interval.
2. Scale of X-axis can either be decided on the basis of class interval or midpoints 3. Join the points plotted for the mid-points corresponding to their frequencies by straight lines. We will get the same figure as obtained by the first method (i.e., with histogram). ■BJ Hr Mui-pomts 15
5
25
12
35
15
45
22
55
14
65
4
Solution.
No. ofstfj^nirw 1
frequency polygon Y Scale : 1 cm = 10 Marks on X-axis 1 cm = 4 Students on V-axis Fig. 13 120 Illustration 7 V
St'^t'stics for Economics-XI
exa JSr
-ks secured by 25 s,ude„,s i„ an
2.1
-
in
- - " " - - - - « . « ,, Solution. Frequency Distribution of Marks Tally Bars Gi foi Marks 20-29 30-39 40-49 50-59 60-69 m( Total -^xuxc preparing 1 exclusive method, i.e.. No. of Students (f) 2 58 64 25 ~1MS Students (f) 19.5-29.5 29.5-39.5 39.5-A9.5 49.5-59.5 59.6-69.5 2 5 8 6 4 J--'-WUTtlUN Scale : 1 cm = lo Marks on *-axis 1 cm = 1 Student on V-axis » i* marks Fig. 14
121 Graphic Fresentation lllusmtion 8. We have the following data on the daily expendttute on food (in rupees) fot 30 households in^alocaU^: ^^^ - - s r/o r/s r/o .s (a) Obtain a frequency distribution using class intervals : 100-150, 150-200, 200250, 250-300 and 300-350 (b) Draw a frequency polygon. ju r-^nt \c) What per cent of the households spend less than Rs 250 per day, and what per cent spend more than Rs 200 per month? Solution, (a) lit Monthly Expenditure on Food (Rs) 100-150
nil
4
150-200
mil
6
200-250
mim^iii
250-300
M
5
300-350
11
A.
Total
30
Tally Bars
13
(b) frequency polygon Scale : 1 cm = Rs. 50 on X-axis 1 cm = 2 Households on V-axis *■ X 100 150 200 250 300 350 EXPENDITURE IN RUPEES Fig. 15 400 122
No. of households (f) . -
fc) Ont of ^sn J, u Statistics for Economics-X ^^
Hence 76.6% spend less tha.
xtlr '' ^^
spend more than
(d) Frequency Curve or Smoothed Frequency Curve generalVby^'el^^^^^^^^^
frequency curve. It is drawn
area mcluded ,s ,ust the same as^I Tthf poL^^^^ "u ^ ^^^^ ^^^"he required to be done carefully to ge^ co rect "eS Smoothing the frequency polygon shows neither more nor less area of the rectanLs of .h v
drawn with care
frequency polygon to get a smoothed freq^ ett
"
histogram for the data given in IllustraZ 8 fo^ U"
^^ constructing
frequency curve Scale : 1 cm = Rs. 50 on X-axis 1 cm = 2 Households on V-axis 200 250 300 expenditure in rupees Fig. 16 >X 350 400 We observe that : 123 Graphic Presentation statistics. It is a uni-modal distribution curve. histogram, frequency polygon and frequency curve _____1—------ I _____.ri^ol 124 ■JJ
Statistics for Economics-XI
ie) y-Shaped Curve (Curve E) : In this case, maximum frequency'is at the ends of rh. (e) Cumulative Frequency Curve (Ogive)
I-
O" the graph paper by rwo
(a)
'Less than' method
(b)
'More than' method
Illustration^^Draw^^ for the following data : Marks 0-10 10-20 20-30 30-40 Ma. of StHdents 44 7 10 Marks 40-50 50-60 60-70 No. of Students 12 obra,„,„g mar,.
S" ^^^^^^^^
125 Graphic Presentation of each class .g in above illustration, the number of students obtain,ng marks more In 0 .s 50; moi; than 10 is 46; more than 20 is 42; and so on. Cumulative Frequency Distribution Marks Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 No. of Students (c.f-) 4 8 15 25 37 45 50
Marks More More More More More More More than 0 than 10 than 20 than 30 than 40 than 50 than 60 Nr. of Students (c.f.) 50 46 42 35 25 13 5 We get a rising curve in than method', if the above case of 'less than method' and declining curve in case of 'more cuLlative frequencies are plotted on the graph paper. ""Ilet the cumulative frequencies of the given frequencies either by 'less than method' or 'more than method'. 2 X-axis — the variables under study 3.
Y-axis - calculated cumulative frequencies
4.
Plot the various points and )om them to get a curve (i.e., ugiv
5.
be clearly lebelled and the scale of the measurement should
be clearly shown. „ ,
Cumulative
'Ogive on Graph Paper
(Cumulative Frequency Curve) by 'less than' method Scale : 1 cm = 10 Marks on X-axis 1 cm = 10 Students on V-axis by Scale more than' method 1 cm = 10 Marks on X-axis 1 cm = 10 Students on V-axis >X 10 20 30 40 50 60 70 80 MARKS 10 20 30 40 50 60 70 MARKS
Fig. 18 Fig. 19
126
■JJ
Statistics for Economics-XI
FAv by less than' method Scale : i cm = 10 Marks on X-axis 1 cm = 10 Students on V-axis Less than method
Fig. 20 ^
c„„e for .he ,oHo„i„, dis„,h.„o„ of
Weekly Wages
Workers (f)
100-109
7
110-119
/ 13 15 32 20 8
120-129 130-139 140-149 150-159 Method 1. Must the lower and upper of he classes. >jet cumulative frequencies Both the axes should be iX l beM T clearly shown. and
«ale of the measurement should be
127 Graphic Presentation Solution, Adjustment of da- limits and calculation of cumulative frequencies by less than method. 99.5-109.5 109.5-119.5 119.5-129.5 129:5-139.5 139.5-149.5 149.5-159.5 7 13 15 32 20 8 7 20 35 67 87 95 ogive (less than method) ocale ■ 1 cm = Rs. 10 on X-axis • 1 cm = 20 Workers on V-axis be ..........Fig. 21
J
and indicate the value o^Ae^i^ ----------128 Solution. Statistics for Economics-XI It I Marks r\ r- Number of Students Cumulative Frequency (Less than) c.f Cumulative Frequency (More than) c.f 0-5 5-10 10-15 15-20 20-25 25-30 30^35 35^0 1 7 10 20 13 12 10 14 9 ----- 1 7 17 37 50 62 72 86 95 88 78 58 45 33 23 9 than' ogive Scale . cm = 5 Marks on X-axis 1 cm = 20 Students on /-axis or h
Kg. 22 Graphic Presentation 129 ^ graphs of time series , ^ , , ....... Time series can be sbown on the graph paper. The information arranged over a period of time (e.g., years, months, weeks, days etc.) is termed as a time series. Presentation of this type of information by hne or curve on the graph paper is of great use in economic statistics. These graphs are known as hne grapjhs or histograms, or arithmetic hne graph. (a) General Rules to Construct a Line Graph 1. As the time (year, month, week) is never in negative (i.e., in minus figures), there is no need of using Quadrant II and III. 2. Year, month or week according to the problem, is taken on X-axis. Give titles to X-axis and Y-axis. 3. Start Y-axis with zero and decide the scales for both the axes. For example, on every 1 cm for Y-axis one may represent an equal gap of 50 students and 1 cm for X-axis a gap between 2000 arfd 2001. X-axis can start either from 1999 or 2000 (See Fig. 23). 4. The pair values will give different dots on the graph paper. For example, values corresponding to time factor are : Years Students 2000 50 2001 150 2002 100 2003 150 2004 200 2005 225 2006 200 These dots obtained of pair values are joined by straight line which is called line graph or histogram (See Fig. 23). students (2000-06)
Scale : 1 cm = 50 Students on V-axis 300250CO H Z 111 200Q Z) H OT
150-
O d
100-
Z 500-
/ /
N
s
f
/ S
/ /
2000 2001 2002 2003 2004 YEARS Fig. 23 2005 2006
130 5. It is not advisable to
■
by a straight iine and not by a curte
"
-is r ^^^^ of un. m One Variable Graph Kendriya Vidyalaya ■ Method 1998-99 1999-00
^^e given below •
J™" Ae dots
2000-01 2001-02 2002-03 2003-04 2004-05 120 400 567 490 760 834 750 Gra Wh in v (yea timt orig largi < year Y-as smal line ; two reqa porti line, I 1 Select X-ax for the time factor (years). 2.
Select Y-ax. for variables under study (students)
3.
G„ble title and scales to X-J Ld C
=
^o Its value and .. the. by
STUDENTS-KENDRIYA VIDYALAYA (1998^)5) Scale : 1 cm = 200 Students on V-axis z UJ Q =3 Ico u. od Sc
2001-02 2002-03 2003-04 2004-05' YEARS Fig. 24 J Graphic Presentation ■ What is a False Base Line? 131 't rlvldTuse faUe base U„e according to ne^ of tbe ptoblent. Keeping doln^—^ .o out tequiretnents by using False ^se Line) mustrafon No. n-bet of stude^ ts ^ one t^usaud .n e.b wmmmmM P" : . . C
T in crranh r nresentation See Fig.
Dortion of the scale may De omiueu wm^n ^ciix ---- , £ that is the use of False Base Line in graphic presentation (See Fig. 25). illustration 13. Present the following information on the graph paper. Year__________
Students
2U00 1120 2001 1380 2002 . 1587 2003 1490 2004 1760 2005 1734 2006 1675 STUDENT&-Cdle : 1 cm = 2 per cent growth rate in years — -----Services Agriculture and allied sectors Industry YEARS Fig. 26 133 IGraphic Presentation 1(d) Graphs of Different Units
different units, we will have two different scales.
When two values are given into two ^^"erent unn , ^^ ^^^ Year 1997-98 1998-99 1999-00 2000-01 2001-02 2002-03
Quantity (in, '000 tons) 9 10 12 11 14 15 Value (Rs. in crores) 300 596 782 900 762 640 Solution. Average of Quantity: 12 Approximately Average of Value : 695 Approximately trade of tea in quantity and value (1997-2003) qcale • 1 cm = Quantity 3 thousand Tons or7-axis : 1 cm = value in RS. 150 crore Quantity -S-Rupees 1997-98 134 Two figures of graphic presentation 10000 exports (Provisional) (US $ IMillicn) Statistics for Economics-Xl\ are shown below to understand time series graph. ^ IMPORTS (Provisional) (US $ Million) 5000 Fig. 28 m Questions :
■
TSL^ Name ehe ,«e.e„.
^ - - .apb . prepared. 7.
berween -Bar dra^ar^'irH™
JO- Wha, « a 'Cumulat,;
^^
Give iUusrrarion.
! r Tr' '"O'-ency curve fop rXt"iara presented .„ rbe «*en an the class intervals h^trtheTa.t^X''""'" pS -
f the ess than type- „„ve and
s.gn,f,cance of Rs 715 f^r the gtven « of da™ "
'"-e
13. Sm Vim Graphic Presentation
^^^
What is a false base hne? Under what conditions would its use be desirable? What is meant by (a) Histogram, and (b) Ogive? Explain their construction with the help of sketches. Distinguish Histogram and Historigram clearly with illustrations. What is a smoothed frequency curve? Discuss briefly various types of frequency curves. Explain the importance of graphic presentation of data. 19. Describe the procedure of drawing histogram when class intervals are (i) equal, and (ii) unequal. i4. 1516. 17. 18. Probl^^s : ^ The frequency distribution of marks obtained by students in a class test is given 40-50 3 below: . Marks
: 0-10 10-20 20-30 30^0
No. of Students : 3 10
14
10
Dtaw a histogram to represent the frequency distribution of marks. Comment on the shape of the histogram. What is histogram? Present the data given in the table below in the form of a Histogram: Mid-points : 115 125 135 145 155 165 175 Frequency : 6 25 48 72 116 60 38 3/ Make a frequency Polygon and Histogram using the given data / Marks Obtained : 10-20 20-30 30-40 40-50 /Number of Students :
5
12
4. Draw Histogram from the following data : r Marks Obtained Number of Students :
6
: 10-20 20-30
10
In a certain colony a'sample of 40 households was selected. The data on daily income for this sample are given as follows : 200
120 350 550 400 140 35085
200 15 30-40 15 22 40-50 10 185 22 50-60 14 50-70 6 195 3 70-80 4 70-100 3 5. 180 170 210 430
110 90 185 140 110 170 250 200 600 800 120 400 350 190 180 200 500 700 350 400 450 630 110, 210 170 250 300 (a)
Construct a Histogram and a frequency polygon.
(b) Show that the area under the polygon is equal to the area under the histogram. (Hint. Get a frequency distribution table to obtain a continuous series). 136 ■JJ
Statistics for Economics-XI
Frequency - f s ''''' 'st: it: S... 15-19, 20-24TQ (b) What percent of th^ hr. u ij ' Size of classes t'::;
-- -- ao-z^
Students
. 30-3.35-.„
^ 10 15
"'Z; ^ Workers : 9 12 15 Weekly Wages of
B cia .
; 'tr '"^r ".^r cZ''^"" Frequency : 4
^ ^^ 1^-24 24-30 30-36
u.
I
. 0-.
3.30 3..0-...0 .0.0 .O-.O
Companies : 2
3
j
^— a-OOO.o^, , 35 3. 3. .0 « ^ ^ iwi
■ 137 Waphic Presentation I P^ .e foUo™. -a —^^^^^^^^^ tr r " \ Profit (Rs in^^
65
80
95
f
i
graph from the
Ps "
--- --------------------------Im-hnrtv
t
Year 'T99"o-91 1991-92 1992-93 1993-94 1994-95 1995-96 1996-97 ,,
12 » 25 31 29 27 35
U. 'p:;:::^ — co. a„a .ot. proauct.o„ of a scooter ma— ; company. Year ^ Production f? (in units) 1 Total Cost I (Rs in lakh) 2001 8500 24 2001 2003 2004 2005 9990 117001330015600
29
34
45
49
., h - E three values of the formula are known, the third can be calculated"
_ J^X Tf ,
X = or IX = NX
if any two of the X■,
^ ' ^^^^^--—--—1 X
10
30
20
30
30
30
40
30
50
30
i:x = 150
150
Measures of Central Tendency 163
150 = 30 N NX = SX 5 X 30 = 150 150 = 150. ^ This property has great utihty in calculatton of wage bills, e.g., average wage Rs 120. No. of workers 2000. ... Total wage bill = N. X = 120 x 2000 = Rs 2,40,000. The relation NX = ZX can be easily used for correcting the value of mean, which is explained in the following illustration. ^ m„s„ado„ 17. -n-e arithmetic mean of a series of 40 as Rs 265. Bnt while calculating ii an item Rs 115 was misread as Rs ISO. Fmd the correct arithmetic mean.
Solution. Since, _ EX X = N EX = NX Here,
X = 265, N = 40
EX = 40 X 265 = 10600 Calculated EX, i.., 10600, is wrong as the us get correct EX by subtracting the incorrect item and adding the correct item Incorrect EX = 10600 Less : Incorrect item
1^0
10450 Add : Correct item Correct
^^X = 10565
Hence, corrected arithmetic mean = Rs 264.12. 40 Illustration 18. Tie arithmetic mean of a given set of data (i) in terms of rupees, and (it) in terms of paise.
164 f Solution. Since Statistics for Economics-XI j
Here, N NX -■"''e. of observations,
Calculated ZX i c Tnn values. Let us cor^t M u 1, ^ value = 25). ^ Incorrect
« correct H (5 observati^s x Rs f^st T
^
^dd : Corrected balance of 5 observations (5x5) -25 Correct es
—----
.
Corrected ZX 525 ^ ^ . N " - — = Rs 105 («) Corrected mean expenditure in terms of n ' Rs 105 X 100 = 10500
^^
= paise 10500. ™ean of 5 items (1, 2 3 4 5 . • . • * ' 4, 5,) is 3, I.e., 1+2+3+4+S EX IS value, sa. 2. we get tbe Alisi ( ll requ ] £ Lo to be < examp class M 2.1 Exa 1 2 3 4 __^
"" "" observations are Rs 5 less ,
IX = IS X = 3 X+2 3 4 5 6 25 5 X~2 -1 0 +1 +2 +3 Xx2 Column 1 : X = 3 (+) Column 2 :X - i: a j j j (X) Column = 2) = 1 ^ = 6 Multiplied 2 = (3 X 2) = 6 51 2 4 6 8 10 30 6 We a JVIaifc Studo Cunn before ap Thus Marks Stud^ 165 ,
after aadition. s.b„action and .uUip^on by
-b»caon and *a.on by *e same constant to their means. Miscellaneous Problems
. , ■ u .wUrr^f^uc mean These problems are
re^r rb^^e rS^^e mean.
1. In the case of open-end classes Example Marks Below 10 10-15 15-20 20-25 Above 25 No. of Students 58 3 4 5 Lower limit of the first class and upp. W o^^ j^^r^^L^niS^^ to be defined by marking an —and last classes. Thus, first example, the same class mterva IS decided M., 5) class would be 5-10 and last class 25-30. 2.
In
the
case of
cumulative frequency
distribution
Example Less than 10 Less than 15 Less than 20 Less than 25 Less than 30 5-10 5 5-15 13 5-20 16 5-25 20 5-30 25 We are given Marks : ^tive frequencies ar;:equired to be converted into ascending class frequencies Thus we get, ^^^^ Marks : 5-10 1(^5 ^^^^^^ Smdents :
5
=4 25-30 (25-10) = 5
=8
-^
^^0-16)
■tr m 166 Example : Statistics for Economics-X, We are given, Marks : Students Marks No. of Students More than 5 More than 10 More than 15 More than 20 More than 25
25 20 12 9 5
5-30 25 10-30 20 15-30 12 20-30 9 25-30 5 .ue^citTef: "r
"
above descending cL„,a.ive
frequencies. They are : Marks : 5-1 o (25-20) Students : =5 10-15 (20-12) = 8 15-20 (12-9) = 3 25-30 20-25 (9-5) =4
5
Merits and Demerits of Arithmetic Mean Merits : Arithmetic Mean ,s the most popularly used because of the following merits ■ . 1. It IS simple to understand and easy to calculate. " 2. It is based on all the observations of the series Therefore it i.' th representative measure. ^neretore, it is the most 3. bias
Its values is always definite. It is rigidly defined and not affected by personal
4.
The calculation of arithmetic mean does not require any specific algement of
^iS m SS^~ - - ^-re mathematical 7 It is fluctuations of sampling and ensues .ability m calculations. /. tt IS a good base for comparison. n«h''o7o?:fcuI onTj^^^i^^^^^^^^ '^e average and be used with caution
mean should
ffc^o 000 a " General Manager's sala;, in a firn, rnfpe^f Sloor"^'
-P'ovee-y clerk Rs jjoo, typiS^Rs
167 Measures of Central Tendency „ ^ Rs on 000+ Rs .S.500 + Rs 4,500+ Rs 2,000 _ g oqO per The average salary will be--4 month. Average calculation is not - "presenmive^ I. is affected by an extreme value of Rs 20,000 paid to the General Managet; , 4 Arithmetic mean can be a value that doe. not extst m the senes at all, ..g., the average of 4, 8 and 9 is = 7, which is not an item of the series. 5 Arithmetic mean gives more impot«,nce to the bigger items and less importance to titr^cflXaecided inst by observa.on. It needs mathematica. calculations. (B) Weighted Arithmetic Average or Weighted Mean 1.
Meaning
2.
Uses of Weighted Mean
3.
Calculation of Weighted Mean
(a)
Equal Weights
(b)
Unequal Weights
arithmetic mean gives equal importance tt/aif the ^^^^^ fact, thL are number One item may be more impot^mt hi—rlorl^^^^^^^^
to different
^o^ as jeights. ,n other
wir«=ights are figures to indicate d>e relative mtportauce of ,tems. 1, « » "ve Cual weigh, to different categories of employees in a fa«o^ 2. WeStled mean is used for comparison of the results of two or more un,verstt,es or boards. 3.
,.u^
It is used' to calculate standardised birth rate and death rate.
, 4. It is used in the construction of Index Numbers.
168 Statistics for Economics XI Calculation of Weighted Mean The formula for calculating weighted arithmetic mean is as under : IWX Xw = where.
Xw = Weighted Arithmetic Mean W= Weights X = The variables Steps. (/) Multiply weights by X and obtain WX Hi) Divide the total (ZWX) by total weights (XW) Solution. of
payment of wage per hour by three ways
Simple Arithmetic Mean (x) : Workers Man Woman Child 8 6 4 ZX=18 EX X=
N 8+6+4 18 : ■: 3 . = Rs 6 per hour. (a) Weighted Mean (Equal Weights) : (Xit;) terSn' - - — -- ana •n Type SWlfittlBil^Bl®
Wages (Rs) ■ X
Workers W
Man Woman Child 8X, 6 X, 4X3 50 Wj 50 50 W3 I.W= 150
400 300 X^W^ 200 X3 W3
IWX = 900
-—-—-'ires of Central Tendency 169 Xw X,w, + XM+2W
- =Rs6
LW 150 Weighted Mean is Rs 6.
.n
Thus, weighted arithmetic mean will be equal to the simple arithmetic mean, when all : items are given equal weights. Xw = X Rs 6 = Rs 6. lb) Weighted Mean (Unequal Weights) : (X«^) Suppose men, women and child workers are 10, 20 and 50 respectively then our ---------Vorkers lyfJtr 'x
W
- V - --_U-^^^---^
Man
8
10
80
Woman
6
20
Child 4
50
200
l.W = 80
120
ZWX = 400
- SWX 400 Xw = = Rs 5 LW 80 Thus, the weighted arithmetic mean will be less than the simple arithmetic mean when items of small vflues are given greater weights and items of big values are given less weights.
__
Xw < X Rs 5 < Rs 6 . u However, in the absence of given weights, assumed weights can be assigned to the items on the basis of their relative importance. But, normally they are not equal. Suppose men, women and child workers are 50, 20 and 10 respectively, then our answer would be different. Type Man Woman Child X Workers W 50 20 10 SW = 80 WX 400 120 40
ZWX = 560 __ ZWX Xw =
560 80 =7 Weighted Mean is Rs 7.
^
Statistics for Economics-Xl
Thus the weighted arithmetic mean will be greater than the simple arithmetic mean when items of small values are given less weights and items of big values are given mor^ weights. Xw > X Rs 7 > Rs 6 niustration 20. Calculate Weighted Mean by weighting each price by the quantity consumed. Articles of
Quantity Consumed
Food
(per kg) 3
Price in Rs
Flour 11.50 5.8 Ghee 5.60
58.4
Sugar .28
8.2
Potato.16
2.5
Oil
20.0
.35
Solution. fiiBsBslPlSSaHi^B^^^^B Food Articles kg W Flour 5.8
11.50
Ghee 58.4
5.60
Sugar 8.2
.28
Price in Rs per kg
Qty. Consumed in
Potato2.5
.16
Oil
.35
20.0
Total
17.89
WX iMM 66.700 327.040 •2.296 0.400 7.000 IWX = 403.436 .Xw = ZWX 403.436 = 22.55 I.W 17.89 Weighted Mean Price is Rs 22.55. lUustration 21. From the results of the two schools A and B given below, state which or them is better.' Oass IX X XI XII Total School .A Appeared 30 50 200 120 400 Passed 25 45 150 75 295 School B
Appeared 100 120 100 80 400 Passed 80 95 70 50 295
171 Measures of Central Tendency ntf Use Weighted Anthmetk Mean after obtaining homogeneous figures, converting into percentages. School A Class Appeared w Passed
Pass % X
IX X XI XII 30 50 200 120 15000 7500
25 45 150 75
LW - 400
WX 8.33 90 75 62.50
2499 4500
LWX = 29499
School B Class Appeared W Passed
Pass % X
IX X XI XII 5000
80 95 70 50 80 79.2 70-62.5
100 120 100 80
XW = 400 School A :
WX 8000 9504 7000
EWX = 29504
LWX 29499 _ 73 75 Xw - 400
_ SWX _ School B : Xw School B is better. 29504 400 = 73.76 m„s«a»„ 22. An exannnaaon was held to decide the award of a -hola*
Subject Statistics Accountancy Economics Business Studies Weight 4 32 i Marks of A 63 65 58 70 Marks of B
JAarks of C
60 64 56 80 65 70 63 52 DUSIIICSS jiuun^o _-________ Of Ihe candidate gening the highest marks .s to be awarded the scholarsWp, who should get it? 172 Statistics for Economics-XI Solution. Subfect W
Weight
Marks of A
Marks ofB
X,
WX,
4
63
252
60
240
65
260
Accountancy 3
65
195
64
192
70
210
Economics
58
116
56
112
63
126
Business Studies
1
70
70
80
80
52
Total IW = 10
EWXj = 633 SWX^ = 624 EWX3 = 648
Statistics
EW
10
2:WX2 LW
2
10
624
Marks of C
WX,
52
EWX3 648 ^t Vt ./v
O^ I
Weighted Mean of B, Xw^ = ^^^
= 62.4 Marks.
Weighted Mean of C, Xw^ = ^^^ =
Marks.
The weighted Mean of C is the highest, hence he is entitled for scholarship. OF FORMULAE AND ABBREVIJmONS [Arithmetic Mean, Properties and Weighted Mean] Type of Series 1. Individual Observations (Ungrouped data) Direct Method - ZX N Short-cut Method Sfi^ Deviation Mjethod X = A+'^f'xC N 2. Discrete Series (Grouped data)
^N
X-A + ^^f xC N
3. Continuous Series (Grouped data)
- -Lfm ^ N
X=A+N
Mathematical Properties of Arithmetic Mean Here, X-X=x Mathematically. (1) 2.(X X) = 0 ljc = 0 If(X - X )- 0 Ifx = 0 , Properties of ■ Arithmetic Mean (3) E(X - X )Ms the least, i.e., Ix^ is minimum ZWX Weighted. Mean Xw = ^^ - N1X1+N2X2 Similarly - N1X1+N2X2 + N3X3 N1 + N2 + N3 (4) NX = IX ii-
Measures of Central Tendency 173 Abbreviations X
= Arithmetic Mean.d =
X-A, i.e., deviations of X
variable from x=
The variables.
zx-
Sum of all the items of the variable X.
an assumed mean. U=
Sum of the deviations of X variabic i
taken from an assumed mean. |
N=
Number of observations or (Z/). C =
f=
Frequency. d' =
Common factor.
X — A , i.e., step deviations of C
E/X = Sum of the product of variable (X)
X-variable from assumed mean
and the frequencies (/). m=
and divided by common factor.
Mid-values. Yd' = Sum of step deviations.
lfm = Sum of the products of mid-points frequencies and the frequencies. A=
Assumed mean
Lfd = Sum of the product of
and their respective deviations.
lLfd' =Sum of the product of deviations
and their respec ive step deviations. X-X = X, i.e., deviations of X variable from the mean. =
Combined mean of two groups. (X- Xf =
X', i.e., square of the deviations
of X variable from mean. Xi-
: Arithmetic Mean of the first group.
Xw = : Weighted Arithmetic Mean.
X, -
: Arithmetic Mean of second group.
W=
N.
: Number of observations in the first
zwx = : Sum of the product of variable
1
group.
N, =
: Number of observations in the
: Weights.
X and weight.
second group. EXERCISES Questions : 1.
What is a statistical average? Mention different types of averages.
2. What are the functions of an average? Discuss the characteristics of good average. Which of the average possesses most of these characteristics? 3. What is meant by 'Central Tendency'? Discuss the essentials of a measure of central tendency. 4.
Name the commonly used measure of central tendency.
174 StattsUcs for Economics-XI
5.
Define the mean. Also explain properties of mean.
6. Why xs arithmetic mean is the most commonly used measure of central tendency^ Llutionr ""
"""
^^ ^ - fr^q-ncy
-- - ^ -sure '
^of the va^es of the vanable from
in
^
"
unweighted mean?
lU. What are the uses of weighted mean? II.. Write notes on (a) Central Tendency, and (b) Weighted Mean. Problems : 1.
Calculate arithmetic averages of the following information :
(a)
Marks obtained by 10 students : 30, 62, 47, 25, 52, 39, 56, 66, 12, 24
(b) Income of 7 families (In Rs) : Also show 590, 575
= o 550, 490, 670, 890, 435,
ic) Height of 8 students (In cm) : 140, 145, 147, 152, 148, 144, 150, 151 2.
.u______________________• " :
Name of batsman Matt 1 ■ Inning
ch I
II Inning
ABCD 60 40 100 20 42 40 80 140 52 ,,
,
^ = 600, c = 147.12 cm.]
Match II I Inning
6
irmx 65 15
I Inning
7
8
9
12 15 28 20 14
expenses of following 10 firms
..^,.
^
II Inning
10 36 18 84 100 70 100
••^ = 42.67, B = 56, C= 62.67 and D = 47.33]
Frequency : 6 ^ fI^J^/"^
11 Inning
20 50 10 40 26 60 8 46
Calculate mean of the following series • 5
Mutch m
10 5 [X= 7.06] Firms Sales (Rs in '000) Expenses (Rs in '000) 1 50 11 2 50 13 3 55 14 4 60 16 5 65 16 7 65 15 8 60 14 9 60 13 10 50 13 [X= Sales = Rs 58,000, Expenses = Rs 14,000] ieasures of Central Tendency Calculate mean of the following frequency distribution 62 64 67 70 73 82 103 176 212 180 Values Frequency 60 54 77 115 n Calculate arithmetic mean of the followmg data Profit (in Rs) : 0-10 No of shops : 12
18
2/
30-40 20 175 81 85 89 78 _ 50 21 [X: R> 70.94]
10-20 20-30
40-50 50-60 16 [ X = Rs 30.45] Compare the average age of mal^injhe_ two countries :_____________________ population of U.K. Age Group 0-5 5-10 10-15 15-20 20-25 25-30 30^0 40-50 50-60 60-65 (in lakhs) 214 258 222 157 145 161 267 184 120 100 18 19 20 18 16 14 27 25 19 17 8. [Average Age India = 25.25 years and UK = 29.404 years] Calculate simple and weighted ar^hmetic averages of the folbwing items : Items Weights Items Weights 68 1 124 9 85 46 128 14 101 31 143 2 102 1 146 4 108 11 151 6 110 7 153 5
112 23 172 2 113 17 [Simple Mean = 121.07 and Weighted Mean = 108.71) Marks No. of Students Less than 10 Less than 20 Less than 30 Lesi than 40 Less than 50 -:100 _
5 15 55 75
A = OU iViaiivai
Also get 5:f(X-X) = 0
176 11. 12. 13. 14. 15. Statistics for EconomicsThere are two branches of an establishment employmg 100 and 80 nerso, rrS
monthly salrfes by thrtwo b^rh
are Rs 27^ and 225 respectively, fmd out the arithmetic mean of the salaries the employees of the establishment as a whole. [Combined Mean = Xi,2 = Rs 252.81 to be"49 Tth'by a group of 100 students were found If Too ; H
obtained in the same examination by another group
of 200 students were 52.32. Fmd out the mean of marks obtained by both th g^ups of students taken together. ^ ^^ 3, The mean marks of 1 >0 students were found to be 40. Later on it was discovere — Ll ' ^^
—^^ -- correspondingTot
-ru
•,
[Correct X = 39.7 Marksl
The mean weight ot 25 boys in group A of a class is 61 kg and the mean weiS of .5 boys m group of the same class is 58 kg Find the mean weight of 60 b^! .. , ,
.
[Xj.2 = 59.25 kg]
Calculate mean ot the following data : Marks Beloiv : 10 20 30 No. of Students : 5 9 17 40 29 50 45 60 60 16. Calculate Combined Mean 70 80 90 100 j 70_ 78 83 85 [X = 48.41 Marks] Section
Mean Marks No. of Students
A
75
50
B
60
60
C
55
50
17
,■
[Xi,2,3 = 63.125 Marks]
average ot 31 marks. What were the average marks of the other students.' -ru
[X2 = 57.25 Marks]!
R^18r4"T
1000 workers of a factory was found to be
taken af29ranri67" 'T?"'
^^ workers were wron^
taken as 297 and 165 mstead of 197 and 185. Find the correct mean.
'
[Corrected Mean : Rs 180.32]! 19. Find the average wage of a worker from the following data • : Above 300 310 320 330 340 350 360 3701 No. of ivorkers : 650 500 425 375 300 275_ 250 100
[X = Rs 339.23]j Measures of Central Tendency 177 effip* ........ ^
o. of day
-40 to -30
10
-30 to -20
28
-20 to -10
30
-10 to 0
42
0 to 10
65
10 to 20
180
20 to 30
10
[X = 4.29 °C) 21. A candidate obtains tbe followmg percentage of marks : Sanskrit Mathemat^ 84, Economics 56, English 78, Politics 57, History 54, Geography 47. ^ is agreed to give double weights to marks m Enghsh, Mathematics and Sanskrit. What is he weighted and simple arithmetic mean?= 68.8, X = 64.43 Marks] 22.
Calculate weighted mean by weighting each price by the quantity consumed:
Food items Flour Ghee Sugar Potato Oil Quantity Consumed 500 kg 200 kg 30 kg 15 kg 40 kg Price in Rupees {per kg) 1.25 20.00 4.50 0.50 5.50
[Xw = Rs 6.35] 23. Comment on the performance of the students of three universities given below using weighted mean : No. of Students are in hundreds Courses of study
Mumbai
% pass No. of students Np. of students
Kolkata % pass
Cher tnai No. of students
M.A. M.Com. B.A. B.Com. B.Sc. M.Sc. T-l 83 73 74 65 66 4 5 2 3 3 65 60 1 3 6 7 3 7 81 76 74 58 70 73 2 3.5 4.5 2 7 2
% pass S2 76 73 76
[Weighted Mean Mumbai : 72.55, Kolkata : 7U.6 and Chennai : 72.0; Mumbai is better] 24. A distribution consists of three components with total frequencies of 200, 250 and 300 having means of 25, 10 and 15 respectively Find out the mean o^ combined : distribution.
=
Chapter 9 TOOTioMHTnaet ui. rmmmv«jm (a)
Median,
(b)
Partition Values (Quart,les), and
(c)
Mode.
median 1.
Definition
2.
Calculation of Median
3.
Mathematical Properties of Median
4.
Merits and Demerits of Median
Definition
^ev^r shit t s^r z"-- --- - ---e sxss lics-XI Positional Average and Partition Values
201
According to AX. Bowley, "If the number of the group are ranked m order according to the measurement under consideration then the measurement of the number most nearly one half ts the median." ^ ■ According to Secrist, "Median of a series is the value of the ttem actual or estimated tvhen a sertes ts arranged in order of magnitude which divides the distribution into the tivo parts. nL! O™'''''' ;
'uf "sr
147 151 140 Anurag Deven 149 M Suresh 142 At Mayoor 147 AtuI 144 144
""
heights of 7 students in a class.
Satish 145 145 Himankar The first and most important rule for obtaining the median is that the data should be arranged in an ascending (increasing) or descending (decreasing) order. This arrangement facditates locating the central position so that the series may be divided into two parts one less than the central value and the other more than the central value. ' So, we arrange our data in ascending order as follows : 140 142 144 145 147
149 151 mm Deven Mayoor
Satish Himankar AtuI Suresti f ■
Anurag If we arrange the above data in descending order we get : Name of smdents : Anurag Suresh Ami Himankar Satish Mayoor Deven Height (cm) : 151 149 147 145 144 142 140 From this ordering also we observe that 145 cm or value of the 4th item is the median. Calculation of median (a)
Individual observations.
(b)
Discrete series.
(c)
Continuous series.
Median is the central positional average of given data. That is, median has a position more or less at the centre of the values and it divides the series roughly into equal parts. 180 Statistics for Economics-XI {a} Individual Observations meiarhetht. ^
^ impute the
Solution. Name of Students Height (cm.) Anurag
151
Deven140 Suresh
149
Mayoor
142
Atul
147
Satish 144 Himanka--
145
Name of Students Height (cm.)
Deven140 Mayoor
142
Satish 144 Himankar Atul
145
147
Suresh
149
Anurag
151
Steps : 1. The above data must be arranged either in ascending or descending order to get the value of median. Arrange the data in ascending order. \th item 2.
Locate the median by finding the size of
3.
Applying the formula, we get
fN+^^ Me = Size of = Size of fN + V th item Pos But fN h the h 7+1 item
= Size of 4'"' item Median is the Himankar's height, i.e., 145 cm 8th tTm
^^
cm, which will be the
»tn item m the list, and calculate the median height. Solution. When the number of items in an individual series is 3, 5, 7, 9, 11 etc. that 'N+iK ■ th Item will be a whole number. nil 1 is when it is an odd number, the central item, i.e.. -XI the f ositional Average and Partition Values
Igj
But when the number of item in a series is even 2, 4, 6, 8, 10 etc, the central item, /.e., N+V the item will be in fraction. Arranging the data in ascending order including the height of Rajesh, we get et Name of the Students Deven140 Mayoor
142
Satish 144 Himankar AtuI
145
147
Suresh
149
Anurag
151
Rajesh
152
Me = Size of = Size of
Heii^ht (cm.)
fN + lX^ . 2,+ Item Item = Size of 4.5'*' item Medkn is estimated by finding the arithmetic mean of two middle values, i.e., adding the height of Himankar and AtuI and dividing by two. '& Size of 4.5"^ item = item + item 2 145 + 147 292 Median height = 146 cm. Serial No.
Marks Serial No.
Marks Serial No.
1
17
7
41
13
11
2
32
8
32
14
15
3
35
9
11
15
35
4
33
10
18
16
23
5
15
11
20
17
38
6
21
12
22
18
12
Marks
JI 182 Statistics for Economics-XI """sed ,„ an ascending order in the Serml No.
Marks Serial No.
Marks Serial No.
Marks
1 2 3 4 5 6 11 11 12 ' 15 15 17 7 8 9 10 11 12 32 13 14 15 16 17 18 32 33 35 35 38 41 Median = Size of the
18 20 211 22
J 23
item = Size of the 18 + V th = 9.5'^ item The value of 9.5"' item = .Z^lue of the 9"* item + Value of the 10^'' item = 11^.21.5. Hence Median = 21.5 (b) Discrete Series Illustration 4. Calculate median of the followmg distribution : Solution. Marks No. of Students 10
1
20
8
30
16
40
26
50
20
60
16
70
7
80
4
Marks 10 20 30 40 50 60 70 80 No. of Students 1 8 16 26 20 16 7 4 N =99 Ctwtulatiue frequencies c.f 1=1 10 = 2 26 = 2 52 = 2 72 = 2 88 = 2 95 = 2 99 = 2 16 16 16 16 16 16
26 26 26 26 26 20 20 20 20 16 16 16 up to (c) Cc nil the m( 7 7+4 Positional Average and Partition Values 183 Steps : 1.
Arrange the data in ascending or descending order.
2.
Compute cumulative frequencies.
3.
Apply the formula.
Me = Size of fN + n th Item. 4. Median is located at the size of the items in whose cumulative frequency, the value of (N + U th item falls. Median = Size of = Size of (N + l
th 2 (99 + ^^ Item = 50th item Median Marks = 40 Marks. Illustration 5. Find out the value of median from the following data : Daily wages (in Rs) : 100 50 70 110 80 Number of Workers : 15 20 15 18 12 Solution. Wages in
Number of
Ascending Order (Rs)
(f)
(c.f.)
50
20
20
70
15
35
80
. 12
47
100
15
62
110
18
80
Cumulative
Workers
Frequencies
Median is the value of fN + l rso+i^i or th or 40.5'*' item. All items from 35 onwards up to 47 have a value of 80. Thus the median value would be Rs 80. (c) Continuous Series Illustration 6. The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.
Size of Land Holdings
No. of Families
(in acres) Less than 100 100-200
89
200-300
148
300-400
64
400 and above
40
39
J' 184 IvV i Solution. Statistics for Economics-XI Size of Land Holdings (in acres) No. of families (f) frequencies 0-100 40
40
100-200
89
129
200-300
148
111
300-400
64
341
400-500 •
39
380
Steps : 1. Compute less than cumulative frequencies. u th item. Do not use
Less than cumulative
2.
Median item is located by finding out size of
the item in continuous series. 3.
Locate the median group in cumulative frequency column where the size of
fN^'' the item falls. 4. Apply the following formula to calculate the median from located group : —— c.f. Median = /j + - x i where, = Lower limit of median group. c.f = Cumulative frequency of the class preceding the median class. f = Frequency of the median group. I = The class interval of the median group. Calculation of Median Me = size of = size of
2 380 item item = 190^'^ item Median lies in the group 200-300. Applying the formula, we get --cf Me = /j +
:
Xi Ositional Average and Partition Values where, /, = 200, f = 190, c.f. = 129 f = 148, /• = 100 185 Me = 200 + 1^1^x100 = 200 = 200 +
148 61x100 148 241.216 148 •. Median size of land holding = 241.22 acres, (ie 50% of the families are having less than or equal to 241.22 acres of land holdingr^d 50% of famihes are having more than or equal to 241.22 acres of land holdings.) Illustration 7. Calculate median from the following data : Age (in years) 55-60 50-55 45-50 40-45 Number of
Age
Persons
(m years)
(f) '7
35-40
13
30-35
15
25-30
20
20-25
Total Number of Persons (f) 3U 33 28 14 160
«I Note : If the given question is in deseending otdet of values then Wore giving the question, the dafa is Required to arrange ■„ ascending order to calculate less than cumulative frequencies.
..
,
Solution. This question has been solved below after arranging the series m ascending order. "___ Age in years (Ascending order) 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 No. of persons (f) 14 28, 33 30 20 15 13 7 Cumulative frequency-(c.f.) 14 42 75 105 125 140 153 160 186 Statistics for Economics-XI In the above example median is the value of lies in 35^0 class interval. N- , Me = + X i 80-75 f^l
th
or
.1)
ri6o> th I2;
or
or 80''^ item which = 35 = 35 30 5x5 30 X5 = 35 + 0.83 = 35.83 .•• Median Age = 35.83 years Illustration 8. Calculate the median from the following data :
Value Frequency (f)
Value Frequency (f)
Less than 10 4
Less than 50
Less than 20 16
Less than 60 112
Less than 30 40
Less than 70 120
Less than 40 76
Less than 80 125
Solution. If the data are given in the form of cumulative series they have to be converted into simple series in order to find out the frequency of the median class which IS needed m calculation of median. Once it is done that rest of the procedure is the same as in any other continuous series. Value Frequence (f) 0-10
4
Cumulative frequency {c.f.)
4
10-20 12
16
20-30 24
40
30^0 36
76
40-50 20
96
50-60 16
112
60-70 8
120
70-80 5
125
ha' on Middle item is ri25 xth or 62.5* item, which lies in 30-40 group. lics-XI Positional Average and Partition Values 187
ich ^ -c.f. Me = /, + - X i ^^ 62.5-40 = 30 + —trr- X 10 = 30 + 36 22.5x10 36 = 30 + 6.25 Median = 36.25 Illustration 9. Calculate the median from the following data Size
Frequency (f)
More than 50
0
More than 40
40
More than 30
98
More than 20
123
More than 10
165
Solution. e>umulative frequency taoie is oi more man type, in !.u».u eases mc ucua have to be converted into a simple continuous series and median is calculated of ascending order series. ,, be lich me Size
Frequency (f)
10-20 42
42
20-30 25
67
30^0 58
125
40-50 40
165
Cumulative frequency (c.f.)
ri65Y' Middle item is —^ or 82.5* item which lies in 30^0 group. th
eI N Me = /j + - X i ^ 30 , iM^iZ >, 10 = 30 + 58 15.5x10 58 = 30 + 2.67 Median = 32.67 188 Statistics for Economics-XI Illustration 10. Compute median from the following data • MMues : 115 125 135 145 155 165 175 185 195 Frequency : 6 25 48 72 116 60 8 22 3 he'^^rdls^^Z^: Th 7fT ^^^
of ^he class-intervals of a contmuous
trequency distribution The difference between two mid-values is 10 hence 10/2 - 5 i. upper limit ot a class. The classes are thus 110-120 170 l^n ^ ^ 190-200. '
.... and so on up to -——----
Uass-intervals 110-120 120-130 130-140 140-150 150-160 160-170 170-180 180-190 190-200 Total
Frequency 6 25 48 72 116 60 38 22 3
390 _ 6 31 79 151 267 327 365 387 390 The middle item is (390^ th or 195"' item, which lies in the 150-160 group. Me = + -
^
195-151 = 150 + -r^^-X 10 116 = 150 + 44 X 10 116 = 150 + 3.79 Median = 153.79 Illustration 11. if the arithmetic mean of the data given is 28 Find rh. I ^ ■ • frequency, and (b) the median of the series. ^ ^^^ Profit per Retail shop (in Rs) : 0-10 10-20 20-30 Number of Retail shop j2 30-40 27 40-50 17 50-60 6
jg
Positional Average and Partition Values 189 Solution. {a) Calculation of missing frequency. Let the missing frequency of group 30-40 he X. Profit per Retail shop X 0-10
12
5
60
10-20 18
15
270
20-30 27
25
675
30-40 X
35
35X
40-50 17
45
765
50-60 6
55
330
N = 80 + X
Number of retail shops f Mid-point m '
—-----1 fm
Y.fm = 2100 + 35X
Applying formula, we get Ifm X= or 28 = Ifm If " N 2100+ 35X 80 + X 28 X (80 + X) = 2100 + 35 X 2240 + 28 X = 2100 + 35 X 2240 - 2100-= 35 X - 28 X 7X = 140 140 X= 7 = 20 Therefore, the missing frequency is 20. (b) Calculation of median :
Profit (Rs) (X)
Fret, mcHC\ (f)
0-10
12
12
10-20
18
30
20-30
27
57
30-40
20
77
40-50
17
. 94
50-60
6
100
N=
(c-f)
100
The middle item is 100^ or 50th item, which lies in the 20-30 ^group. 190 Statistics for Economics-XI Me = /j + -ly- X -in ^0-30
20x10
= 20 + —-X 10 = 20 + 27 "
■ 27
= 20 + ^ = 20 + 7.407 Median = 27.41. Illustration 12. In the frequency distribution of 100 famiUes given below, the number of families corresponding to expenditure groups 20-40 and 60-80 are missing from the table. However, the median is known to be 50. Find the missing frequencies. Expenditure : 0-20 20-40 40-60 60-80 80-100 No. of families : 14 ? TI 15 Solution. Let the missing trequency of the group 20-40 be X and the missing frequency of 60-80 group be Y. Now Z/" (total frequency) = 100
i.e., 100 = 14 + X + 27 + y + 15 or X + Y = 100 - 14 - 27 - 15 or X + Y = 44 Expenditure No. of Families (f) Cumulative frequency (c.f.) 0-20
14
14
20-40 X
14 + X
40-60 27
41 + X
60-80 Y
41 + X + Y
80-100
15
100
Median is given in this problem as 50. ^lOOV*' Middle item of the series is also interval 40-60. (Given median = 50) N or 50* item, which means it lies in the classNow, Me = /. + -c.f. f Xt 50 = 40 + 50-[14 + X] 17 X 20 50 - 40 = 27^20 f ositional Average and Partition Values
191
10 X 1.35 = 50 - [14 + X] 13.5 = 50 - 14 - X X = 50 - 14 - 13.5 = 22.5 Since the frequency in this problem cannot be in fraction so X, i.e., f^ wouId be taken as 23.
X + Y = 44 or /■j + = 44 /; = 44 - fj or 44 - 23 or 21 Thus the missing frequencies in the question are 23 and 21. Mathematical Properties of Median 1. Median is an average of position and therefore is influenced by the position of items in arrangement and not by the size of items. 2. The sum of the deviations of the items about the median, ignoring ± signs, will be less than any other point. For example : X
: 10 11 12
Deviations from Median : 2
10
Deviations from any poirit, (say 10) :
0
1
2
The sum of the deviations taken from median (12), less than the sum of the deviations taken from an\ 13 1 14 2 = t>
-f — »J ipomt (1®
Merits and Demerits of Median ^^''^i^ilJAN t Merits 1.
It is easy to calculate and understand.
2. It is well defined as an ideal average should be and it indicates the yalue of the middle item in the distribution. 3.
It can be determined graphically, mean cannot be graphically determined.
4. It is proper average for qualitative data where items are not converted or measured but are scored. 5.
It is not affected by extreme values.
6. In the case of open-end distribution it is specially useful since only the position is to be known. It is useful in a distribution of unequal classes. Demerits 1.
For median data need to be arranged in ascending or descending order.
2.
It is not based on all the observations of the series.
3.
It cannot be given further algebraic treatment.
4.
It is affected by fluctuations of sampling.
192 Statistics for Economics-XI 5.
It is not accurate when the data is not large.
6. Interpolation by a formula is required to calculate median in continuous series This reqmres the assumption that all the frequencies of the class interval are uniformly spread which is not always true. partition values (quartiles) 1.
Definition
2.
Characteristics of Partition Values
3.
Calculation of Partition Values
Definition When we are required to divide a series into more than two parts, the dividing places are known as partition values. Suppose, we have a piece of cloth 100 metres long an^d we have to cut it into 4 equal pieces, we will have to cut it at three places. Quartiles are those values which divide the series into four equal parts. For getting partition values the most important rule is that the values must be arranged m ascending order only. In the case of finding out the median, we can arrange the data either m ascending or in descending order but here there is no choice-only ascending order is possible for calculating partition values (Quartiles).
For example, we have the following data of heights of 7 students in a class • Name of students : Anurag Deven Suresh Mayoor Atul Satish Himankar Height (cm) : 151 140 149 142 147 144 145 Therefore, for getting correct results, the data must be arranged in ascending order in all the cases. Characteristic of Partition Values The difference between averages and partition values is as follows : While an average is representative of whole series, quartiles are averages of parts of series For example, the first quartile is the average of first half of the series and third quartile is the average of the second half of the series. Thus, quartiles are not averages like mean and median. They help us in understanding how various "ems are spread around the median. Therefore, the special use of partition values IS to study the dispersion of items in relation to the median, that is in understanding the composition of a series. Calculation of Partition Values (a)
Individual Series.
(b)
Discrete Series.
(c)
Continuous Series.
Positional Average and Partition Values Now we arrange the data (in Illustration 1) in ascending order : 193 Name of Students (cm) Deven140 Mayoor
142 = Q == First quartile or lower quartile
Satish 144 Himankar AtuI
147
145 = Me = = Second quartile or middle quartile
Suresh
149 = Qj = Third quartile or upper quartile
Anurag
151
AS we Know tne meuiaii is uic nci^iii. iwuim ------------cm. Now, suppose we have to calculate quartiles. By definition quartiles will divide a series into four equal parts and so number or quartiles will be three. They are known as lower quartile, middle quartile and upper quartile. These are also called first, second and third quartiles. The middle or second quartile (Q^) is the central positional value of the data, i.e., median. The first or lower quartile (Qj) is the central positional value of the lower half, and third or upper quartile (Q3) is the central position value of upper half of the data. In the above data, (Q, = 142, Q, = 145 and Q, = 149. It must be remembered that Q, is always less than Q^ and Q3 (Q^ < Q^ and Q3) and median falls between Qj and Q3. (a) Individual Series Illustration 13. From the following information of wages of 30 workers in a factory calculate median, lower and upper quartile. S. No. Wages (in Rs) 1
330
16
240
2
320
17
330
3
550
18
420
4
470
19
380
5
210
20
450
6
500
21
260
7
270
22
330
8
120
23
440
9
680
24
480
10
490
25
520 .
11
400
26
300
12
170
27
580
S. No. Wages (in Rs)
13
440
28
370
14
480
29
380
15
620
30
350
194 Solution. 1.
Arrange the data in ascending order.
2.
Locate the item by finding out.
fN+iY^ rN+v""' and N+1 Nth items Median Me = size of N+l 2 th Item r 30 + 1* = size of —J- item = 15.5* item ^ size of 15th item + size of 16th item 380 + 400 Statistics for Economics-XI S. No. *
Wages (in Rs)
1
16
120
400
S. No. Wages (inRs)
2
170
17
420
3 4 5 210
18
440
240
19
440
260
20
450
6
270
21
470
7oi
300 1 ^
22
01
320
23
480
9
330
24
490
10 11 12 13 14 15 330
480
25
330 350 370 380 380
500 26 27 28 29 30
520 550 580 620 680
F I Uj {b)l I Calci = 390 Median is Rs 390. Positional Average and Partition Values Lower Quartile 195 Qj = size of - size of rN+n th item (30 + 1 ah item = 7.75th = size of item + |(size of - size of 7* item)
= 300 + .75 (320 - 300) = 300 + 15 = 315 .-. Lower Quartile is Rs 315. Upper Quartile Qj = size of = size of VN+r"' item V3O + 1Y'' item = size of 23.25* item = size of 23^" item + -^(size of 24* item - size of 23^" item) = 480 + ^(490 - 480) = 480 + .25(10) = 480 = 480 + 2.50 Upper Quartile is Rs 482.50. (&) Discrete Series r ,
uu
Illustration 14. Following are the different sizes and number of shoes m a shoe shop. Calculate median, first quartile and third quartile.
Size of Shoes 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11
No. of Shoes (f) 4 8 12 15 20 35 50 40 20 15 24 12 5 3 196 Solution. Statistics for Economics-XI I Steps Sue of shues
" -----
4.5
4
4
5
8
12
5.5
12
24
6
15
39
6.5
20
59
7
35
94
7.5
50
144
8
40
184
8.5
20
204
9
15
219
9.5
24
243
10
12
255
10.5
5
260
11
3
263
dat: 1.
Arrangement of the data in ascending order is necessary.
2.
Calculate less than cumulative frequencies.
3.
Locate the item by finding out :
fN + U th fN + V th and Af + 1 th Wk " ""
"
feo-cy, the tule of
Median Me - size of = size of fN + lY'' . Item r263+r th item = 132th item First Quartile = size of 132* pair of shoes = 7.5 size of shoes. N + l^ th 4 263 + 1^*'' Qj = size of = size of = size of 66* item = size of 66* pair of shoes Medial Ap: item item lics-XI Positional Average and Partition Values
First Quartile = 7 size of shoes. Third Quartile 197 Q = size of Vn+T''* Item = size of r 263 + 1^ th item = size of 198* item Third Quartile =8.5 size of shoes, niustration 15. Calculate Median, First Quartile and Third Quartile from the following data: Solution. Income
No. of persons
(in Rupees) 800
16
1000 24 1200 26 1400 30 1600 20 1800 5 Income
■ ^H
(in Rs)1 800
16
16
1000 24
40
1200 26
66
1400 30
96
1600 20
116
1800 5
121
Median : Applying formula, we get Me = size of = size of fN + 1 Nth item ri2i+i^ th item = 61* item = income 61* person
198 Median = Rs 1200 First Quartile Statistics for Economics-XI Qj = size of
4 Item = size of 121+n Item = 30.5* item = income 35.5* person ••• Qi = Rs 1,000 Third Quartile Q. = size of = size of
(N+l^ th Item T21 + n th 4 item Thus, = 91.5* item = income 91.5* person Q, = Rs 1,400 Me = Rs 1,200 = Rs 1,000 Q3 = Rs 1,400 (c) Continuous Series Marks Students Solution. Me 30-35 14 35^0 16 40-45 18 45-50 23 50-55 18 55-60 8 60-65 3 Marks 30-35 35-40 40-45 45-50 50-55 55-60 60-65 Mo. of students (f) 14 16 18 23 18 8 3 c.f 14 30 48 71 89 97 100
H lics-XI Positional Average and Partition Values 199 Steps : 1.
Calculate less than cumulative frequencies.
2.
Median, first quartile and third quartile items are located by finding out
th
n/»T\th
u. (N^ v4. , and N Item m continuous series. 3. Locate the median group, first quartile and third quartile group by cumulative frequency column where the size of respective fall. 4. Apply the suitable formula to get the value : Me = /j + - X i Nr — -C.f. 2. th N4 th , and fN' 4,.
th Items
fN) 4] -c.f. XI Median Median = size of-^ item = = 50* item Hence, median lies in class 45-50 N , Me = /j + - X i th 100 where, = 45, ^ = 50, c.f = 48, f ^ 23, / = 5 50-48 Me = 45 + = 45 + 23 2x5 23 X5 = 45.43 Hence, median is 45.4% marks. 200 First Quartile Qj = size of Hence, Q^ lies in class 35-40 ^
Statistics for Economics-Xl
14 j item = = 25* item
—-c.f. XI where, ^ 35, ~= 25, c.f = 14, / = 16, i = 5 = 35 + 1^= 38.43 16 Hence, first quartile is 38.4% marks. Third Quartile Qj = size of Hence, Q lies in class 50-55 (N^
th
rioo^i
Item = ^ UJ
I4J
= 75* item
-c.f. f X/
v4. where, = 50, ' — = 75, c.f = 71, f = 18, / = 5 . ^ „ 75-71 = 50 + = 51.11 lo Hence, third quartile is 51.11% marks.
niustration 17. Calculate the Median and Q^ using the following data : Mid-points marks : 5
15
25
■ No. of students : 3 10
17
35 7 45 6 55 465 2 75 1 Positional Average and Partition Values
201
Solution. Given mid-points are required to be converted into class intervals. 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 3 10 17 76 4 .2 1 Calculation of Median and Q3. Median Applying formula, we get Median = size of th U>
r5o^
item =
I2 J
= 25* item Hence, Median lies in class 20-30 Applying suitable formula to get the median value "l-c-f. Me = /j + ^^— X i where I, = 20,^ = 25, c.f. = 13, f = 17, i = 10
2S-1 3 Me = 20 + - X 10 = 20 17 12x10 Hence, Median is 27.05 marks. Third Quartile 17 = 27.05 n v4y item = ^^ = 37.5* item Qj = size of Hence, Q^ lies in class 40-50 Applying suitable formula, we get /XT / Nl Qs = K + v4. -c.f. f X/ 3 13 30 37 43 ■47 49 50 202 Statistics for Economics-XI
. where, = 40,
= 37.5, c.f = 37, f= 6, i = 10 03 = 40.^^,10 . 40 . = 40.83 Hence, third quartile is 40.83 marks. Illustration 18. Calculate the Median and Quartiles for the following : Marks (below) : 10 20 30 40 50 60 70 80 No. of Students : 15 35 60 84 96 127 198 250 Solution. Before calculating Median and Quartiles, first we convert the given cumulative frequencies into class frequencies : [ W. of^tttdentfi 0-10
15
15
10-20 20
35
20-30 25
60
30-40 24
84
40-50 12
96
50^60
31
60-70 71 ■
198
70-80 52
250
127
Total 250 Median. Applying formula, we get ( N^^ Median = size of v2. 250
,
Item = -— = 125* item Hence, median hes in class 50-60 Applying suitable formula to get median :
Me = /, + 2 f Xt where / = 50, N = 125, c.f = 96, f= 31, i = 10 125-96 2 Me = 50 = 50 + 31 29x10 31 - X 10 = 59.35 ;. '.Hence, median is 59.35 marks. •i Tl in a Positional Average and Partition Values First Quartile rN 2) Nth item = 250^ Q^ = size of Hence, Q, lies in class 30-40. Applying suitable formula, we get E^cf. = 62.5* item
Xt 62.5-60 = 30 +
X 10
. 30 . ^ 31.04 24 Hence, Q, is 31.04 marks. Third Quartile fN th r250 4J Q, = size of -J- item = Hence, Q, lies in class 60-70. Applying suitable formula, we get 4. = 187.5* item Q3 = ^
f XI 203 r" Hence, Q, is 68.52 marks. Thns, Q, = 31.04, Q. = Median = 59.35 and Q, = 68.52 marks. nlustarion 19. The following series relates to the da,ly income of workers employed in a firm. Compute (a)
highest income of lowest 50% workers,
(b)
minimum income earned by the top 25% workers, and
(c)
maximum income earned by lowest 25% workers.
204 Statistics for Economics-XI 25-29 30-34 35-39 20 10 5 Daily Income (in Rs) : 10-14 15-19 20-24 Number of workers :
5
10
15
Before solving it let us understand the question. 1. As the data are of inclusive class intervals, we are required to convert the classes into class boundaries. ^.lasses 2.
The data is arranged in ascending order, where
Area of 50% of workers in highest income group At this point a worker in the centre earning highest daily income of the lowest 50% of workers (;.e., Median value) 3. The data is arranged in ascending order, where - 100% data --------1_ . H-4 At this point a worker is eaming minimum daily income of top 25% workers .4.Area of top 25% of workers 4. The data is arranged in ascending order, where Area of lowest 25% of workers ■ At this point a worker is earning maximum daily income of lowest 25% workers (i.e. lower quartile value = O,) qutdln"'
^^
^^^ ^^^
to the given
Positional Average and Partition Values SoiutLoa. 205 DaUy Income (m (X) 9.5-14.5 14.5-19.5 19.5-24.5 24.5-29.5 29.5-34.5 34.5-39.5 5 10 15 20 10 5 5
15 30 50 60 65 (a) Computation of highest daily income in lowest 50% of workers. (Median) Nth Median is the value of th —
(65\
item or
Uv
-
I2j
or 32.5th item which lies in 24.529.5 class interval. Applying suitable formula to get median value, N Me = /j + -c.f. Xt = 24.5 + = 24.5 + f 32.5-30 20 2.5x5 x5 20 = 24.5 + 0.625 = 25.125 .-. Highest data income of lowest 50% workers is Rs 25.13. (b) Computation of minimum daily income earned by top 25% workers (Q^)
th 3x65 ^ l\ Q, = Value of — item = Hence, Q3 lies in class 24.5 - 29.5. Applying suitable formula, we get
.T, = 48.75* value Qs = ^ -c.f. f Xt 48.75-30 _ = 24.5 + -TT-x^ = 24.5 + 20 18.75x5 20 = 24.5 + 4.687 = 29.187 Minimum daily income earned by top 25% workers is Rs 29.19. It,f Statistics for Economics-XI (c) Computation of maximum daily income earned by lowest 25% workers (Qj) Qi = Value of UJ item = 4r = 16.25* value Hence, Q^ lies in class 19.5 - 24.5 Applying suitable formula, we get Q. = ^ f XI = 19.5 + = 19.5 + 16.25-15 15 1.25x5 15 x5 = 19.5 + 0.416 = 19.916 Maximum daily income earned by lowest 25% workers is Rs 19.92.
Graphical Determination of Median and Quartiles Illustration 20. Determine median and quartiles graphically from the following data : Marks : 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35^0 Students : 7 10 . 20 13 17 10 14 9 Solution. Secc Akrrifes . . f ■ Mjr/fes less than More than cumulative 0-5
7
5
7
0
100
5-10
10
10
17
5
93
10-15 20
15
i7
10
83
15-20 13
20
50
15
63
20-25 17
25
67
20
50
25-30 10
30
77
25
33
30-35 14
35
91
30
23
35^0 9
40
100
35
9
Less than cumulative
Marks more than
N = 100 First Method (only for median). Steps 1. Calculate ascending cumulative frequencies (less than) and descending cumulative frequencies (more than). 2.
Draw two ogives—one by 'less than' and other by 'more than' methods.
3 4. 5. lics-XI Positional Average and Partition Values 3.
From the intersecting point of two ogives, draw a perpendicular on X-axis.
4.
The point where perpendicular touches X-axis, median value is determined.
207
Second Method (For Median and Quartiles). Steps 1.
Calculate ascending cumulative frequency (less than).
2.
Determine the value by the following formulae :
Me = size of Qj = size of Q = size of
th UJ
item, i.e., ^^ = 50* item u Item, i.e.. N4 th Item, i.e.. 100 4 3^100^ = 25* item = 75* item V^/ 3. Locate 50, 25, 75 values on Y-axis and from them draw perpendiculars or cumulative frequency curve (ogive). 4. From these points where they meet the ogive draw another perpendicular touching X-axis. 5.
The points where perpendicular touches X-axis, Qj, Me and Q^ are located.
208 Statistics for Economics-XI
VehficaUon Median Group 15-20 Me = /j + NU -cf f Xt 50-37 ^ 13x5 Median = 20 Marks Lower Quartile Group 10-15
-cf XI 25-17 = 10 + —-X 5 - 10 20 8x5 20 Qi = 12 Marks. = 12 Upper Quartile Group 25 - 30
-cf xt ic 75-67 ^ 8x5
= 25.— .29 Q, = 29 Marks Less than method' cumulative frequency curve is the reminder of the rule that at the hrst step of calculation of quartiles, the data is arranged in ascending order. Howeven median can be ocated on graph even by more than 'ogive' or calculated by arranging the data m descending order. ^hb^ Positional Average and Partition Values 209 1.
Definition
2.
Determination of Mode
3.
Merits and Demerits of Mode
1. Definition According to Coxton and Cowden, "the mode of distribution is the value at the point around which the items tend to be most heavily concentrated. It may be regarded as the most typical of a series of values."
x/r j •
The word mode comes from French la mode which means the fashion Mode in statistical language is that value which occurs most often in a senes, that is value which is most typical. If garment manufacturers say that short collars are now in fashion the statement implies that maximum number of people now-a-days wear short collar shirts If we say the mode is size No. 7 shoe, it means in a given data maximum number of people wear size No. 7. Thus, mode is that value of observations which occurs the greatest number of times or with the greatest frequency. For a better understanding of mode let us look at the following information about frequency of students in relation to marks obtained. Marks : 5 10 15 20 25 30 35 40 45 50 : 2 3 25 2 1 18 20 24 14 10 According to the explanation of mode given above, the modal marks will be 15 because maximum number of students (25) have obtained 15 marks each. Although 15 have the highest frequency, a more careful examination of the information
shows that the highest concentration of the frequency is around 40 marks. That is, m the neighbourhood of 40 marks. There are more frequencies (18, 20, 14, 10) as compared to the neighbourhood of 15 marks (2, 3, 2, 1). Thus 15 marks are not ^yp.c^/ of the series of valLs. For the reasons given above, 40 marks is the mode and not 15. Therefore, to define accurately, mode is that value of observations around which items are most densely or heavily concentrated. The mode is defined as the most frequently occurring value. If each observation occurs the same number of times, then there is no mode in that distribution. If two or more observations occur the same number of times (and more frequently than any other observation) then there is more than one mode and the distribution is multi-modal, as against uni-modal, where there is one mode. If two values occur most frequently then the series is bi-modal, in case of three values occurring most frequently then the series is called tri-modal. The mode as a measure of central tendency has little sigmficance for a bi- or "Mode is that value of the graded quantity at wh,ch the instances are most numerous. " -A.L. Bowley "The value occurring most frequently in a senes (or group) of Hems and around which the other item^ar^ distributed most densely." 210 Statistics for Economics-XI - -
.oae .„.
2. Determination of Mode (a)
Series of Individual Observations and Discrete Series
(b)
Continuous Series
(c)
Graphic Location of Mode
id) Mode from Mean and Median. (a) Series of Individual Observations and Discrete Series In a senes of individual observations, the mode can be located in two ways • "
--
a cl^r"""
^^st ^^ ^^^
Marks : 4 6 5 ' in 98
-rks obtamed by 15 students i
Solution. 10 4 7 6 5 Modal value 8 7 7 7 8 8 9 9 10. (a) (i) Array : 4 4 5 5 6 6 : Mode = 7 Marks («) Discrete Series. Converting the above data into discrete series, we get Mode = 7 Marks lics-XI Positional Average and Partition Values
233
(b) Discrete Series. In discrete series the mode can be located by two ways : (i)
By Inspection.
(ii)
By Grouping.
(i) By Inspection. The mode can be determined just by inspection in discrete series, the size around which the items are most heavily concentrated will be decided as mode. Illustration 22. Find out mode from the following data : Wages (in Rs)___ 125
3-
175
8
225
21
275
6
325
4
375
2
' No.- of Persons ■
Solution. By inspection, we can determine that the modal wage is Rs 225 because this value occurred the maximum number of times, i.e., 21 times. {ii) By Grouping. In discrete and continuous series, if the items are concentrated at more than one value, attempt is made to find out the item of concentration with the help of grouping method. In such situations it is desirable to prepare a grouping table and an analysis table for ascertaining the modal class.
In grouping method, values are first arranged in ascending order and the frequencies against each item are properly written. A grouping table normally consists of six columns Frequencies are added in twos and threes and total are written between the values. It necessary, they can be added in fours and fives also. Column 1. The maximum frequency is observed by putting a mark or a circle. Column 2. Frequencies are grouped in twos. Column 3. Leaving the first frequency, other frequencies are grouped in twos. Column 4. Frequencies are grouped in threes. Column 5. Leaving the first frequency, other frequencies are grouped in threes. Column 6. Leaving the first two frequencies, other frequencies are grouped in threes. After observing maximum total in each of these cases, put a mark or circle on every total. An analysis table is prepared after completing grouping table in order to find out the item which is repeated the highest number of times. If the same procedure is adopted in continuous series, we shall be in a position to determine the modal class. We shall now see how mode is determined by grouping method in a discrete series. 212 Statistics for Economics-XI Illustration 23. Find out mode of a data given in Illustration 20 by grouping. Grouping Table Wages (m Rs) •
• nr^a—ft and
-.an
tendency ■ "raTr^ndelf ^
-d,a„ as measures of
and mode of a frequency " rtrsLr ^^ -- - --/02 = 1.15 ^
+ 0.59 = 7.59
281 N 217 Applying the formula, we get Here, C.V = I X 100 o = 1.15 and X = 7.59 C.V = ^ X 100 = 0.1515 X 100 = 15.15% Continuous Series itmuous series niustration 32. To check the quality of two bulbs and their life in burmng hours was Life (in hrs.) Ho. of bulbs Brand A 0-50
15
Brands
2
50-100
20
8
100-150
18
60
150-200
25
25
200-250
22
5
Total 100
100
(i) Which brand gives higher life? (it) Which brand is more dependable?
282 Solution. Statistics for Eco. nomics-Xl Brand A Coefficient of Variation (Brand A^ C-V- = ^ X 100 Let us calculate first a and X. Standard Deviation MLjlfd"
Xc Here,Z/^- = 193,Z/-^-19,N=100andC=50 CT = /m 1100 '19_ 100 X 50 Co^ffidem ofJS^ation~(BrLd BJ = X 100 Let us calculate first a and X. Standard Deviation a= l^jEfd'^ Xc
a= 100 .100, x50 Measures of Dispersion
283 = Vl.93-(0.19)2 Vl.93-0.0361x50 = Vl.8939x50 = 1.376 X 50 = 68.8 hrs. Arithmetic Mean X=A+
XC
N Here, A = 125, I^d' = l9,h!= 100 and C = 50 19 = Vo.61-(0.23)2 - V0.61-0.0529x50 >/0.5571x50 0.746 X 50 = 37.32 hrs. X = 125 + X 50 Arithmetic Mean Ifd' X=A+
XC
N Here, A = 125, Ifd' = 23, 100 and C = 50 23 100 = 125 + (0.19) X 50 = 125 + 9.5 = 134.5 hrs. Applying the formula, now we get C.V. = ^ X 100 .A. where, a = 68.8 and X = 134.5 68 8
C.V = X 100 = 51.15%. 134.5 X = 125 + X 50 100 = 125 + (0.23) X 50 = 125 + 11.5 = 136.5 hrs. Applying the formula, now we get where. C.V. = ^ X 100 .A. a = 37.32 and X = 136.5 37.32 C.V. = 136.5 X 100 = 27.34%. (/■) Since the average life of bulbs of brand B (136.5 hrs) is greater than that of brand A (134.5 hrs), therefore the bulbs of brand B give a higher life. (ii) Since C.V. of bulbs of brand B (27.34%) is less than that of brand A (51.15%), therefore the bulbs of B are more dependable. Illustration 33. The number of employees, wages per employee and the variance of wages per employee for two factories are given below : No. of Employees Average wage per employee per day (Rs) Variance of wages per employee per day (Rs) (a) In which factory is there greater variation in the distribution of wages per employee? (b) Suppose in factory B, the wages of an employee are wrongly noted as Rs 120 instead of Rs 100. What would be the corrected variance for factory B? Factory A 50
100
Factory B
120
85
9
16
284 Solution. («) Calculation of Coefficient of Variation : Factory A Statistics for Economics-XI C.V. = |. X 100 Here, x = 120 and a = = il? 100 = 2.5% Factory B C.V. = J X 100 Here X = 8S and a = ^ C-V. = ^ X 100 = 4.7% W &rrecti„8 Mean and Variation : — zx For Factory B .^ = 100 and X= 85 100 x 85 = 8500 It IS not correct ZX Corrected ZX= 8500 - 120 . 100 = Rs 8480 Corrected X = ^ = ^ ^ N Variance = a^ a^ = Here, N 100
- (xp = 16, X = 85 and JV = 100 ZX^ 16 = 100 (85)2 T.
= ^^00 + 722500 = 724100
It IS not correct ZX^
/^^lUU
Corrected ZX^ = 724100 - (120)2 . (100)^ = 724100 - 14400 + 100000 = 719700 Corrected Variance = ^o^reaed ZX^ - (Corrected X)^ 719700 ~ ~Tdr - ^^dard deviation for a comLed
Coefficient ofvariation of two series are 5S«/ ' = ",, = 91 and What are their ml, TJ"""'"'on f the coefficient of variation of X series is J4 6»/ 'f f" * = nteans are fOf.a and
respe':,!:^^^.^- l^dTZ:^--
No. Of Persons COOO)
^^
60 iqO i^q
40
50
22.608]
Class A Class B
]'
20
80
^ - li It ll - « « .5 85 „ ' ^ - - « 40 61. xi/St------y^PC-
58. 59. 60. - [rniu-vaiues) No. of students (Eco.) No. of students (Statis.) Chapter 11
MEASURES OF CORRELATION T. 2. 3. 4. 5. 6. Introduction Correlation and Causation Kinds of Correlation Degree of Correlation Methods of Studying Correlation Scatter Diagram Karl Pearson's Coefficient of Correlation (£) Spearman's Rank Correlation List of Formulae
iduction In the previous chapters we have discussed measures of central tendency (Mean, Median and Mode), partitional values (Quartiles) and measures of dispersion (Range,' Quartde Deviation, Mean Deviation, Standard Deviation and Lorenz Curve. These are all relating to the description and analysis of single variable only This type of statistical analysis is called 'univariate analysis'. Now, we will deal with problems involving association in two variables. We find that in social as well as natural sciences, where more than one inter-dependent variables are involved, change in one variable brings change in others. For instance, in Biology we know that weight of a person increases with height in Geometry we know the circumference of a circle depends on the radius, in Economics prices vary with supply, cost of industrial production varies with the cost of raw materialsagricultural production depends on the rainfall etc. The relationship between variables is measured by correlation analysis. Thus, 'the term correlation (or covariation) indicates the . relationship between two such variables in which change in the values of one variable, the values of the other variable also change.' This statistical analysis of such data is called bivariate analysis Other Definitions According to Croxton and Cowden, "When the relationship is of d quantitative nature, the appropriate statistical tool for developing and measuring the relationship and expressing it in a brief formula is known correlation."
According to L.R. Connor, "If two or more quantities vary in sympathy so that movements in one tend to be accompanied by corresponding movements in other(s) then they are said to be correlated." t 314 H. ^relation and causation j„ .
Statistics for Economics-XI
1. Cause and effect : There is a cause and effect relationship between two variables shon w,ves and many .h„„ starred husbands may havt K' ^ r; Measures of Correlation
315
be correlation between price and demand so that in general whenever there is an increase in price the demand falls, and vice-versa. But this does not mean that whenever there is a rise in price the demand must fall. It is possible that with the rise in price the demand may also go up. This is on account of the fact that in economic and social sciences various factors affect the data simultaneously and it is difficult almost impossible to study the effects of these factors separately. Thus, correlation measures co-variation, not causation. It measures the direction and intensity of relationship among variables. s^ds of correlation
On the basis of nature of relationship between the variables correlation may be : (1)
Positive and Negative Correlation
(2)
.Linear and Curvilinear Correlation
(3)
Simple, Multiple and Partial Correlation
1. Positive and Negative Correlation When both the variables change in one direction, that is when both increase or decrease the relationship between the two variables is called positive or direct. But when the change is in opposite directions that is one is increasing and the other is decreasing, the correlation is negative or inverse. For determining the direction of change average values are taken. For example : (I) Positive Correlation
(il) Negative Correlation
(a) .
(b)
(a)
Both variables increasing
(b)
Both variables decreasing
One variable
increasing, the
other decreasing
One variable decreasing, the
other increasing
X
Y
X
Y
X
Y
X
Y
10
100
70 .
147
15
125
75
110
20
150
60
140
30
110
60
180
30
160
40
135
35
90
40
190
40
190
30
130
40
80
30
200
50
200
15
120
45
75
20
240
60
255
10
90
50
60
10
250
We find that in I (a) the values of X series are increasing so also of the Y series. In I (b) values of X and Y are decreasing. Thus, they are both instance and positive correlation. On the other hand, in II {a) the values of X are increasing and the values of Y are decreasing, similarly in II (b) the values of X are decreasing and the values of Y are increasing. Thus, hey are both examples of negative correlation.
316 Examples : (Positive Correlation)
'^"""""'cs-XI
1- Age of husband and age of wife. --easeinheatlllT^^^nofr,. 1- Demand of a commodity mav ^ Increase in the number tfTe ^ "": " 3- Sale of woollen garments ant ly ir "" Yield of crops and Price. " Correlation u the ratio of chance h^n of^rrotstr
-ear
line. If tbTv:ri:trfmS™=™™bles. Their^SaSnshbt'f "P™ ""-"ncy "f non-hnear "ttnTw
are ^'phed, rhe
bear a constant rari^ ('""'''near), the amount of chale 3. Staple, Multiple and Partial Correlation relationship betwln • T
"^"Me or
p ''
^^r-ables are
IS a study of relaf,V,„,l.- 7 ""her variables from ,1. u , influencing valbles b ''
" ™riables
variables Sfllcit ^ '
of tainfalf""stant. For example, 3;,':''
"f other
correlation. " a certain consrant^Jm:: M
Measures of Correlation 317 of correlation The relationship between two values can be determined by the quantitative value of coefficient of correlation which is obtained by calculations. Perfect Correlation : Perfect correlation is that where changes in two related variables are exactly proportional. If equal proportional changes are in the same direction, there is perfect positive correlation betWeen the two values described as +1; and if equal proportional changes are in the reverse direction, there is perfect negative correlation, described as - 1. For example, the circumference of a circle increases in the equal proportionate ratio with the increase in the equal proportionate ratio in the length of its diameter; the amount of electricity bill increase in a perfectly definite ratio with an increase in the number of unit consumed, the volume of a gas varies inversely with the pressure at constant temperature etc. DEGREE OF CORRELATION ive Zero Correlation : The value of the coefficient of correlation may be zero. It means that there is zero correlation. It does not mean the absence of any type of relation between the two variables. Two valued are uncorrelated. However; other type of relation may be there. There is no linear relationship between them.
Limited Degree of Correlation : In social science, the variables may be correlated, but an increase in one variable need not always be accompanied by a corresponding or equal increase (or decrease) in the other variable. Correlation is said to be limited positive when there are unequal changes in the two variables in the same direction; and correlation is limited negative when there are unequal changes in the reverse direction. The limited degree of combination can be high (between ± .75 to 1); moderate (± .25 to .75) or low
i j I-i
318 (between H-0 tn a u- u
Statistics for Economics-Xl
Karl Pearson's foLt '"P""'^-"'^ degree of eorrelarion according ,o Dejp-ee of Correlation Perfect Correlation Very high degree of Correlation Sufficiently high degree of Correlation Moderate degree of Correlation Only the possibility of a Correlation Possibly no Correlation Zero Correlation (Uncorrelated) Positive +1 + -9 or more from + .75 to + .9 from + .6 to +.75 from + .3 ro +.6 Less than +.3 0 Negative -1 - .9 or more from - .75 to - .9 from - .6 to -.75 from - .3 to -.6 Less than -.3 0 IhhoD; "on. Somell^ are :
SW
Coefficient of Correlation
{c) Spearman's Rank Correlation
scatter diagram
tdea about the presence of measuring X-variable on bokolTL^^apb paper. The chart is prepared by pomt for each pair of observation oTx and y JZTi u'''' ^^ P^^^ plotted m the shape of points. The cluster of ooin^ U the scatter diagram. When the plottedTo „te^
^^^^^ ^ata are
"" P^P^^ called
we know that there is some correlation Seen tl ^^^d-upward or downward-the correlat^n is positive, when it ir^ov^nw^d ^ - "Pward - scatter diagrams given below : ^"rreiation is negative. Let us study r=+l Perfect Positive Correlation (a) High Degree ot Positive Correlation (b) Low Degree of Positive Correlation (0 Measures of Correlation Perfect Negative Correlation (d) 319 High Dgree of Negative Correlation Low Degree of Negative Correlation I
w
V!
i
r= 0
No Con-elation (9) Fig. 1 Figure [a), (b) and (c) show an upward trend—they show positive correlation. Figure (d), (e) and show a downward trend—they show negative correlat!on. Howe\ er, there are differences among (a), (b) and (c) and similar differences among {d), (e) and (/). We find from the plottings on the scatter diagrams that there is a certain similarity among (a) and {d), (b) and (e) and (c) and (/). In (a) and (d) the plotted points are almost in a straight lines—this indicates perfect correlation. In [b) and (e) the plotted points are not in a straight line but if we draw a straight line in the middle of their points (regression line) we will find, the points are near about the line. This kind of scatter diagram shows high degree correlation. In (c) and (f) if we draw a similar line (regression line), we will find that the plotted points are very much scattered around the line—^not as near as in the case of {b) and (e). This kind of scattered diagram shows low degree correlation. Finally, diagram {g) shows such a vast scatter of points that it is impossible to see any trend— this shows no correlation or zero correlation. Illustration 1. From the following pairs of value of variables X and Y draw a scatter diagram and interpret the result. 8 9 10 11 12 13 14 15 54 48 42 36 30 24 18 12 5 72 6 66 7 60 X : 4 Y : 78 Solution. We note that X = 4 and Y = 78 as given first X and Y values. We may plot this as point (X, Y) on graph paper, where X = 4 and Y = 78. We measure 4 on X-axis and 78 lik ,1. j„ .
Statistics for Economics-XI
scatter diagram Scale : 0.5 cm = 2 on X-axis
B 64 56 48 40 32 24 16 8 0
320 coordinates of measure 5 along the x'axis and 72 alongT axis and so on for all d,e given X and y Xl from the above scatter diagram we can decide Aat the variables X an" Y "e corre ated. The points take the shape of li^e hen tC r 'r "" Weei X aid Rate of Change It is slope of the straight line rwhirh depends on an angle that the str^lghT Hnt makes with the X-axis and is equal to j^Zj rate of change showing almost equal change t-- » .__
—
—
i•
—
—
--
1 • -4-.
-U
—1-
02468 10 12 14 16 IS showing more than proportionate
----
an< change I'fl:: m in
a (i Measures of Correlation 321 showing less than proportionate change
showing no change non-linear relationship Fig. 3 We know when the plotted points show some upward trend, the correlation is positive and when there is downward trend, the correlation is negative. (/•) If the straight line makes an angle of 45° with the X-axis, the change is exactly in the same proportion as the change in the value of X [Fig. {a) and (b)]. Hi) If the angle that the straight line makes with the X-axis is greater than 45° the change in the value Y is more than proportionate to the change in the value of X [Fig. (c) and (d)]. (iii) If the angle that the straight line makes with X-axis is less than 45", the change in value Y is less than proportionate to the change in the value X [Fig. (e) and (/)] (w) If there is no angle and it is a straight line parallel to X-axis, it shows that value Y does not change at all [Fig. (v) Linear correlation exists when the ratio of change between two variables is uniform The relationship is described by the straight line. In case of non-linear relationship (curvilinear) the amount of change in one variable does not bear a constant ratio
to the amount of change in the other variable. Such relationship will form a curve on graph [Fig. (h)]. 322 Ir ! m
»f Statistics for Economics-XI Merits and Demerits of Scatter Diagram Merits : 1. It is very easy to draw a scatter diagram. 5. fa case of linear relationship between x lni y lT T ^ donate change in th^ .a,„e t Jcha^^r,: t'^nTT" " t?™ r^atX™ — ■'"own ,„ n„„er,ca, Whether YLTes XorTcau^ —^^ ^oes not tell, ^hen ,t . not possible to draw a scatter pearson's coefficient of correlation X fgr^BS
variable
(1867-1936).ItisthemostLerused me^^^^^^^^^ and correlation of coefficient. This is atrcaHedX^^^^
statistician Karl Pearson
represented by r. It is based on arithmel^ a„d f't'!, ^^ ^ of Correlation (r) of two variables « ob aiZ l jfjS^^T,^-^ffft-ent corresponding deviations of the various 7eZ of the products of the by the product of their standard devZ^Zan JZ ^T Symbolically, r= Ixy
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