Lesson 1-05 Measuring Central Tendency STAT.docx

Chapter 1: Describing Data Lesson 5: Measuring Measurin g Central Tendency Tendency TIME FRAME: 1 hour session OVERVIEW OF LESSON

The lesson begins with students engaging in a review of various measures of central tendency. ollowing the review! review! students are given cases where these measures measures are calculated. "tudents are also as#ed to e$amine both strengths strengths and limitations of these measures. "ome time will be devoted to having students discuss %uestions with a p artner before reporting to the class. &ssessments will be given to students on their ability to calculate these measures! an d also to get an overall sense of whether they recogni'e how these measures responds to change s in data values. LEARNING OUTCOME(S): &t the end of the lesson! the learner is is able to • • •

calculate commonly used measures of central tendenc y!  provide a sound interpretation of these summary measures! and discuss the limitations of these measures.

LESSON OUTLINE:

1. (ntr (ntrod oduct uctio ion) n)* *arm +p ,. Case Case "tud "tudie iess -. &nalys &nalysis is and Commen Comments ts on Case Casess DEVELOPMENT OF THE LESSON

&/(ntroduction)*arm +p or 10 minutes! let students recall that data h as variation as# them what would be some so me ways to describe the center of a data set2 Three commonly used measures of the center are: • • •

Mean Median Mode

(nform students that the most widely used measure of the center is the arithmetic/ mean. 3

The mean of a data set is the sum of the data values divided by the number of data values.

Chapter 1 Describing Data – Lesson 5

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& basic feature of the mean! also called the average! is the ease in calculation. &ll the data contribute e%ually in its calculation. That is! the 4weight of each of the data items in the list is the reciprocal of the number 6 of data! i.e. 1)6. Mention to students that the mean represents the 4center of gravity. That is! if the values in a list were to be put on a dot scale! the mean acts as the balancing point where smaller  observations will 4balance the larger ones. "pecial 6ote: & measure of economic performance called the 7ross Domestic 8roduct 7D8/! which represents the value of all goods and services produced within the domestic territory for a specific period of time. The 7D8 can also be related with the goods and services which go to consumption! to investments! including those that go to e$ports less the country9s imports. *hen 7D8 is divided by total population! we have some average measure of income or e$penditure in the domestic territory. *hen a country9s economic production and growth as measured by the 7D8/ is healthy! we e$pect to see low unemployment and increases in incomes as businesses demand more labor to meet the growing economy. &n abrupt change in the 7D8 also has effects on the stoc# mar#et. & healthy economy which indicates high consumption and production would translate into higher profits for  companies! which in turn! would increase stoc# prices. *hen there are e$tremes in a set of data! the mean is not be a good measure of the center. ne alternative measure of the center is the median! the cut off where the data are split evenly into lows and highs.

The median of a data set is the 4middle observation when the data set is sorted in either increasing or decreasing order/. 6ote that when the si'e n is even! the median is the average of the two middle scores.

(nform students that the median is fairly easy to calculate particularly when the si'e of the data is rather small. ;owever! for moderate and large data sets! the median may be tedious to compute! as sorting the data would be cumbersome. *ith available computing tools such as spreadsheet applications li#e andomly select a group to present their group wor# for 5 to 10 minutes! with the remaining groups as#ed to ma#e comments for - to 5 minutes on the presentations. Case 1: &veraging (ncomes There are -? families living in your neighborhood. The household family monthly incomes are given in the following table: , families @ ?0!000 - families @ ,0!000 B families @ -,!,50  families @ ,5!000

5 families @ -A!000 ? families @ ,?!000 , families @ A0!000 1 family @ 1,!000

Last wee#! one mansion at the end of the street was ust finished being built! and the family of Manny 8ac%uiao decided to move inE "uppose that the monthly income of the 8ac%uiaos is 1-.AF million pesos.

&s# students to ma#e a histogram to represent the new household incomes for their street. rom the histogram! as# them to estimate where the center is. Tell them to calculate the Mean! Median! and Mode for the income data! with and without the 8ac%uiao family. (nstruct students to write a paragraph e$plaining what the best choice for the measure of central tendency is! and why.

Chapter 1 Describing Data – Lesson 5

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"pecial 6otes: i/

ii/

(nform students that the 8ac%uiao income is called an 4outlier! i.e. a value in a dataset that is not very typical in relation to the rest of the data/. These outliers seriously affect the mean! but not the median no r the mode. Mean! Median! and Mode without 8ac%uiao are -0!5-A.FA?F1! -,!,50 and -,!,50! respectively while Mean! Median! and Mode with 8ac%uiao are ?,0!,-5.F1?-! -,,50! and -,,50! respectively. Thus! students should indicate that the median is the  best choice for an average! when we consider income distribution. The mean gets easily affected by the presence of the e$treme observation the high income of the 8ac%uiao family/! increasing the average from about -1 thousand pesos to over ?00 hundred thousand.

Case ,: Color for the "enior ;igh "chool Dance or the senior high school dance! there is a debate going on among students regarding the color that will be featured prominently. Gotes were sent by students via "M"! and the results are as follows: >ed H -00 votes

Iellow H ,,0

7reen H 550 votes

=lue H F10

range H F0 votes.

=rown H -5

*hite H 1-0 votes

8urple H 5

&s# students to ma#e a 8ie 7raph showing the outcome of the election. Tell students to identify if there is a clear winner o n the choice of color. (nstruct students to find the Mean! Median! and Mode for the colors! if possible not the amount of votesE/ Tell them to write a paragraph e$plaining why you could or could not find each measure of the center. *hich measure of center will determine the color to be prominently used during the senior high school dance2 Case -: >esults of Jui' in "tatistics and 8robability Course . 7. ,00/.=asic "tatistics for the Tertiary Level ed. >oberto 8adua! *elfredo 8atungan! 6elia Mar%ue'/! published by >e$ =oo#store.

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ASSESSMENT

1/ Thirty people were as#ed! 4how many people do you consider your best friend. The graph  below shows their responses. *hat measure of center would you use to find the center for the number of best friends people have2