1 Descriptive Statistics 2-1 Overview 2-2 Summarizing Data with Frequency Tables 2-3 Pictures of Data 2-4 Measures of Center 2-5 Measures of Variation 2-6 Measures of Position 2-7 Exploratory Data Analysis (EDA)

2  Descriptive Statistics summarize or describe the important characteristics of a known set of population data  Inferential Statistics use sample data to make inferences (or generalizations) about a population 2 -1 Overview

3 1. Center: A representative or average value that indicates where the middle of the data set is located 2. Variation: A measure of the amount that the values vary among themselves 3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed) 4. Outliers: Sample values that lie very far away from the vast majority of other sample values 5. Time: Changing characteristics of the data over time Important Characteristics of Data

4  Frequency Table lists classes (or categories) of values, along with frequencies (or counts) of the number of values that fall into each class 2-2 Summarizing Data With Frequency Tables

5 Qwerty Keyboard Word Ratings Table

6 Frequency Table of Qwerty Word Ratings Table RatingFrequency

7 Frequency Table Definitions

8 Lower Class Limits Lower Class Limits RatingFrequency are the smallest numbers that can actually belong to different classes

9 Upper Class Limits Upper Class Limits RatingFrequency are the largest numbers that can actually belong to different classes

10 are the numbers used to separate classes, but without the gaps created by class limits Class Boundaries

11 number separating classes Class Boundaries RatingFrequency

12 Class Boundaries Class Boundaries RatingFrequency number separating classes

13 midpoints of the classes Class Midpoints

14 midpoints of the classes Class Midpoints Class Midpoints RatingFrequency

15 is the difference between two consecutive lower class limits or two consecutive class boundaries Class Width

16 Class Width RatingFrequency is the difference between two consecutive lower class limits or two consecutive class boundaries

17 1. Be sure that the classes are mutually exclusive. 2. Include all classes, even if the frequency is zero. 3. Try to use the same width for all classes. 4. Select convenient numbers for class limits. 5. Use between 5 and 20 classes. 6. The sum of the class frequencies must equal the number of original data values. Guidelines For Frequency Tables

18 3. Select for the first lower limit either the lowest score or a convenient value slightly less than the lowest score. 4. Add the class width to the starting point to get the second lower class limit, add the width to the second lower limit to get the third, and so on. 5. List the lower class limits in a vertical column and enter the upper class limits. 6. Represent each score by a tally mark in the appropriate class. Total tally marks to find the total frequency for each class. Constructing A Frequency Table 1. Decide on the number of classes. 2. Determine the class width by dividing the range by the number of classes (range = highest score - lowest score) and round up. class width  round up of range number of classes

19

20 Relative Frequency Table relative frequency = class frequency sum of all frequencies

21 Relative Frequency Table RatingFrequency % % % % % Rating Relative Frequency 20/52 = 38.5% 14/52 = 26.9% etc. Total frequency = 52

22 Cumulative Frequency Table Cumulative Frequencies RatingFrequency Less than 320 Less than 634 Less than 949 Less than Less than Rating Cumulative Frequency

23 Frequency Tables Rating Frequency % % % % % Rating Relative Frequency Less than 3 20 Less than 6 34 Less than 9 49 Less than Less than Rating Cumulative Frequency

24 a value at the center or middle of a data set Measures of Center

25 Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values Definitions

26 Notation  denotes the addition of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population

27 Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x x

28 Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population is pronounced ‘x-bar’ and denotes the mean of a set of sample values Calculators can calculate the mean of data x = n  x x x N µ =  x x

29 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

30 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’) ~

31 Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’)  is not affected by an extreme value ~

no exact middle -- shared by two numbers (even number of values) MEDIAN is 5.02

(in order - odd number of values) exact middle MEDIAN is no exact middle -- shared by two numbers (even number of values) MEDIAN is 5.02

34 Definitions  Mode the score that occurs most frequently Bimodal Multimodal No Mode denoted by M the only measure of central tendency that can be used with nominal data

35 a b c Examples  Mode is 5  Bimodal - 2 and 6  No Mode

36  Midrange the value midway between the highest and lowest values in the original data set Definitions

37  Midrange the value midway between the highest and lowest values in the original data set Definitions Midrange = highest score + lowest score 2

38  Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half.  Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. Definitions

39 Skewness Mode = Mean = Median SYMMETRIC

40 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median

41 Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median

42 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence

43 Jefferson Valley Bank Bank of Providence Jefferson Valley Bank Bank of Providence Mean Median Mode Midrange Waiting Times of Bank Customers at Different Banks in minutes

44 Dotplots of Waiting Times

45 Measures of Variation

46 Measures of Variation Range value highest lowest value

47 a measure of variation of the scores about the mean (average deviation from the mean) Measures of Variation Standard Deviation

48 Sample Standard Deviation Formula calculators can compute the sample standard deviation of data  ( x - x ) 2 n - 1 S =S =

49 Sample Standard Deviation Shortcut Formula n ( n - 1) s = n (  x 2 ) - (  x ) 2 calculators can compute the sample standard deviation of data

50  x - x Mean Absolute Deviation Formula n

51 Population Standard Deviation calculators can compute the population standard deviation of data 2  ( x - µ ) N  =

52 Measures of Variation Variance standard deviation squared

53 Measures of Variation Variance standard deviation squared s  2 2 } use square key on calculator Notation

54 Sample Variance Population Variance  ( x - x ) 2 n - 1 s 2 =  (x - µ)2 (x - µ)2 N  2 =

55 Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value)

56 Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s  = highest value - lowest value 4

57 x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

58 x - s x x + sx + s 68% within 1 standard deviation 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

59 x - 2s x - s x x + 2s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% FIGURE 2-15

60 x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell-shaped distributions ) 0.1% 2.4% 13.5% FIGURE 2-15

61 Chebyshev’s Theorem  applies to distributions of any shape.  the proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 - 1/K 2, where K is any positive number greater than 1.  at least 3/4 (75%) of all values lie within 2 standard deviations of the mean.  at least 8/9 (89%) of all values lie within 3 standard deviations of the mean.

62 Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations.

63  z Score (or standard score) the number of standard deviations that a given value x is above or below the mean Measures of Position

64 Sample z = x - x s Population z = x - µ  Measures of Position z score

Z Unusual Values Unusual Values Ordinary Values Interpreting Z Scores FIGURE 2-16

66 Measures of Position Quartiles, Deciles, Percentiles

67 Q 1, Q 2, Q 3 divides ranked scores into four equal parts Quartiles 25% Q3Q3 Q2Q2 Q1Q1 (minimum)(maximum) (median)

68 D 1, D 2, D 3, D 4, D 5, D 6, D 7, D 8, D 9 divides ranked data into ten equal parts Deciles 10% D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D 9

69 99 Percentiles Percentiles

70 Quartiles, Deciles, Percentiles Fractiles (Quantiles) partitions data into approximately equal parts

71 Exploratory Data Analysis the process of using statistical tools (such as graphs, measures of center, and measures of variation) to investigate the data sets in order to understand their important characteristics

72 Outliers  a value located very far away from almost all of the other values  an extreme value  can have a dramatic effect on the mean, standard deviation, and on the scale of the histogram so that the true nature of the distribution is totally obscured

73 Boxplots ( Box-and-Whisker Diagram) Reveals the:  center of the data  spread of the data  distribution of the data  presence of outliers Excellent for comparing two or more data sets

74 Boxplots 5 - number summary  Minimum  first quartile Q 1  Median (Q 2 )  third quartile Q 3  Maximum

75 Boxplots Boxplot of Qwerty Word Ratings

76 Bell-ShapedSkewed Boxplots Uniform

77 Exploring  Measures of center: mean, median, and mode  Measures of variation: Standard deviation and range  Measures of spread and relative location: minimum values, maximum value, and quartiles  Unusual values: outliers  Distribution: histograms, stem-leaf plots, and boxplots