Mathematics_Notes 2016 HSC

MATHS

NOTES | KEVIN LUU | 5.2 – 5.3 MATHEMATICS

STATISTICS

ASSESSMENT  Review types of data, collecting data, sorting data, measures of central tendency, measures of spread and displaying data.  Interpreting results from two sets of data (i.e. back to back stem and leaf displays, histograms, double column graphs, or box and whiskers plots).  Find the range, interquartile range and standard deviation as measures of spread of data sets - Find the mean and standard deviation of a set of data using digital technologies – calculators - Compare and describe the spread of sets of data with the same mean but different standard deviations  Bivariate Data: recognises the difference between dependent and independent variables. Describes the strength and direction of the relationship between two variables displayed in a scatter plot, e.g. Strong positive relationships, weak negative relationships with justifications.  Uses lines of best fit to predict what might happen between known data values (interpolation) and predict what might happen beyond known data values (extrapolation).  Know the six processes to setting up statistical investigations.  Identify reasons why data in a display may be misrepresented. EXPRECTATIONS  Use measures of central tendency (mean, mode, median) and the range to analyse data that is displayed in a frequency table, stemand-leaf plot or dot plot.  Use terms ‘skewed’ or ‘symmetrical’ when describing the shape of a distribution.  Compare two sets of data and draw conclusions by finding the mean, mode and/ or median, and the range of both sets.  Construct a cumulative frequency table, histogram and polygon (ogive) for ungrouped data.  Use cumulative frequency to find the median.  Group data into class intervals.  Construct a cumulative frequency table and histogram for grouped data.  Find the mean and modal class of grouped data.  Determine the upper and lower quartiles for a set of scores.  Construct a box-and-whisker plot using the five-point summary.  Use a calculator to find the standard deviation of a set of scores.  Use the mean and standard deviation to compare two sets of data.  Compare the relative merits of measures of spread (range, interquartile range and standard deviation). STATISTIC TERMANOLOGY BIVARIATE DATA - data that has to variables

BOX PLOT (CAT-AND-WHISKERS PLOT) - a diagram obtained from the five number summary - the box shows the middle 50% of scores (the interquartile range) - the whiskers show us the extent of the bottom and top quartiles as well as the range CENSUS - a survey of a whole population CUMULATIVE FREQUENCY - the number of scores less than or equal to a particular outcome - e.g. For the data 3,6,5,3,5,5,4,3,3,6 the cumulative frequency of 5 is 8 (there are 8 scores of 5 or less) CUMULATIVE FREQUENCY HISTORGRAM (AND POLYGON) - these show the outcomes and their cumulative frequencies DATA - the pieces of information (or ‘scores’) to be examined - categorical: data that uses non-numerical categories - ordered data involves a ranking, e.g. exam grades, garment sizes - distinct data has no order, e.g. colours, types of cars - numerical: data that uses numbers to show ‘how much’ - continuous data can have any numerical value within a range, e.g. height - discrete data is restricted to certain numerical values, e.g. number of pets DOT PLOT - a type of graph that uses one axis and a number of dots above the axis EXTRAPOLATION - predicting a data beyond the range of values given FIVE NUMBER SUMMARY - a set of numbers consisting of the minimum score, the three quartiles and the maximum score FREQUENCY - the number of times an outcome occurs in the data - e.g. for the data 3,6,5,3,5,5,4,3,3,6 the outcome 5 has a frequency of 3 FREQUENCY DISTRIBUTION TABLE - a table that shows all the possible outcomes and their frequencies (it usually is extended by adding other columns such as the cumulative frequency) FREQUENCY HISTROGRAM - a type of column graph showing the outcomes and their frequencies. FREQUENCY POLYGON - a type of line graph showing outcomes and their frequencies - to complete the polygon, the outcomes immediately above and below the actual outcomes are used (the height of these columns is zero) GROUPED DATA

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data that is organised into groups or classes class intervals: the size of the groups into which the data is organised e.g. 1-5 (5 scores); 11-20 (10 scores) - class centre: the middle outcome of a class e.g. the class 1-5 has a class centre of 3 INTERPOLATION - estimating data that lie within the domain of the values given INTERQUARTILE RANGE - the range of the middle 50% of scores - the difference between the median of the upper half of scores and the median of the lower half of scores - IQR = Q3-Q1 LINE OF BEST FIT - a line that ‘best fits; the data on a scatter plot mean MEAN - the number obtained by ‘evening out’ all the scores until they are equal - e.g. if the scores 3,6,5,3,5,5,4,3,3,6 were ‘evened out’, the number obtained would be 4.3 - to obtain the mean, we divide the sum of the scores with the total number of scores MEDIAN - the middle score for an odd number of scores or the mean of the middle two scores for an even number of scores - the median class is grouped data containing the median MODE (MODAL CLASS) - the outcome or class that contains the most scores OGIVE - this is another name for the cumulative frequency polygon OUTCOME - a possible value of the data OUTLIER - a score that is separated from the main body of scores QUARTILES - the points that divide the scores the scores up into quarters - the second quartile, Q2, divides the scores into halves (Q2 = median) - the first quartile, Q1, is the median of the lower half of scores - the third quartile, Q3, is the median of the upper half of scores RANGE - the difference between the highest and lowest scores SAMPLE - a part (usually a small part) of a large population - random sample: a sample taken so that each member of the population has the same change of being included - systematic sample: a sample selected according to some ordering scheme, e.g. every tenth member - stratified sample: a sample is proportionally taken from each subgroup in a population

SCATTER PLOT - a graph that uses points on a number plane to show the relationship between two categories. SHAPE (OF A DISTRIBUTION) - a set of scores can be symmetrical or skewed SOURCES OF DATA - primary: the data has been collected by yourself - secondary: the data has come from an external source, e.g. newspapers, internet STANDARD DEVIATION - a measure of spread that can be thought of as the average distance of scores from the mean - the larger the standard deviation, the larger the spread STATISTICS - the collection, organisation and interpretation of numerical data STEM-AND-LEAF PLOT - a graph that shows the spread of scores without losing the identity of the data - ordered stem-and-leaf plot: the leaves are placed in order - back-to-back stem-and-leaf plot: this can be used to compare two sets of scores, one set on each side VARIABLE - something that can be observed, measured or counted to provide data

1 STATISTICS TYPES OF DATA The data we collect is made up of variables. These are pieces of information like a quantity or a characteristic that can be observed or measured. They may change either over time or between individual observations. The main types of data are: CATEGORICAL – VARIABLES ARE CATEGORIES - ordered | e.g. exam grades, garment sizes - distinct | e.g. types of cars, eye colour NUMERICAL – VARIABLES ARE NUMBERS - discrete | e.g. goals scored, number of pets

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continuous | e.g. height of a person, distance thrown

COLLECTING DATA There are three main ways of collecting data, including: CENSUS - a whole population is surveyed, e.g. every student in the school is questioned SAMPLE - a selected group of a population is surveyed, e.g. a small number in each class is questioned OBSERVATION - numerical facts are collected and tabulated, e.g. sports data, weather, sales figures, etc. A sample is usually random to limit the chances of bias occurring. However, it may be systematic if the members of the sample are chosen according to a rule, such as every 10th member of a population. If a population is composed of various sub-groups, the sample could be stratified to ensure a proportionate representation of each group in the sample. Primary source data is collected first hand by observation or survey. Secondary source data is obtained from an external source such as a newspaper, website or another person’s research.

SORTING DATA A large amount of data needs to be tabulated (organised into a table) so that it can be analysed. A common form of table is the frequency distribution table. DISCRETE DATA OUTCOME ( TALLY x ) 1 ||| 2 |||| 3 ||||||| 4 ||||||||| 5 ||||| 6 ||

FREQUENCY ( f ) 3 4 7 9 5 2 TOTAL | 30

f ×x

CUMULATIVE FREQUENCY 3 7 14 23 28 30

3 8 21 36 25 12 | 105

GROUPED DATA Used to cluster discrete data into groups or to divide continuous data into adjoining groups. f × c . c . CUMULATIV CLASS CLASS TALLY FREQUENCY

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