Chapter 3. Descriptive Measures. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 1

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1 Chapter 3 Descriptive Measures Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 1

2 Chapter 3 Descriptive Measures Mean, Median and Mode Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 2

3 Section 3.1 Measures of Center Where is the sample distribution centered? Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 3

4 Figure 2.6 (Histogram Review) Cutpoint grouping. Weight of 18- to 24-year old males: (a) frequency histogram; (b) relative-frequency histogram Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 4

5 High and Low Temperatures in 71 US Cities (one-year) (chapter review problem 33, page 91) Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 5

6 Example 3.1: Weekly Salaries: Small Consulting Company wanted to compare the salaries of employees in the first half of summer to those placed in the second half of the Summer. Data Set I Data Set II Stem-and-leaf of Salary1 N = 13, Leaf Unit = (4) Stem-and-leaf of Salary2 N = 10, Leaf Unit = Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 6

7 Definition 3.1 Mean of a Data Set The mean of a data set is the sum of the observations divided by the number of observations. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 7

8 Example 3.1: Weekly Salaries: Small Consulting Company wanted to compare the salaries of employees in the first half of summer to those placed in the second half of the Summer. Data Set I Order Statistics Stem-and-leaf of SALARY N = 13, Leaf Unit = 10 Stem-and-leaf of SALARY N = 13, Leaf Unit = (4) Order Leaves (4) Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 8

9 Example 3.1: Weekly Salaries: Small Consulting Company wanted to compare the salaries of employees in the first half of summer to those placed in the second half of the Summer. Data Set I Order Statistics Stem-and-leaf of SALARY N = 13, Leaf Unit = (4) Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 9

10 Example 3.1: Weekly Salaries: Small Consulting Company wanted to compare the salaries of employees in the first half of summer to those placed in the second half of the Summer. Data Set II Order Statistics Stem-and-leaf of Salary2 N = 10, Leaf Unit = Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 10

11 Definition 3.4 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 11

12 Example 3.1: Weekly Salaries: Small Consulting Company wanted to compare the salaries of employees in the first half of summer to those placed in the second half of the Summer. Data Set I Data Set II Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 12

13 Definition 3.2 Median of a Data Set Arrange the data in increasing order. If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list. If the number of observations is even, then the median is the mean of the two middle observations in the ordered list. In both cases, if we let n denote the number of observations, then the median is at position (n + 1) / 2 in the ordered list. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 13

14 Definition 3.3 Mode of a Data Set Find the frequency of each value in the data set. If no value occurs more than once, then the data set has no mode. Otherwise, any value that occurs with the greatest frequency is a mode of the data set. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 14

15 Example 3.1: Weekly Salaries: Small Consulting Company wanted to compare the salaries of employees in the first half of summer to those placed in the second half of the Summer. Data Set I Data Set II so, 5 th and 6 th values Set I: th value Set II: th value 6 th value Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 15

16 Tables 3.1, 3.2 & 3.4 Data Set I Data Set II Means, medians, and modes of salaries in Data Set I and Data Set II Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 16

17 Figure 3.1 Relative positions of the mean and median for (a) right-skewed, (b) symmetric, and (c) left-skewed distributions Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 17

18 Section 3.2 Measures of Variation Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 18

19 Figure 3.2 Five starting players on two basketball teams Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 19

20 Figure 3.3 Shortest and tallest starting players on the teams Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 20

21 Definition 3.5 Range of a Data Set The range of a data set is given by the formula Range = Max Min, where Max and Min denote the maximum and minimum observations, respectively. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 21

22 Definition 3.6 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 22

23 Key Fact 3.1 Variation and the Standard Deviation The more variation that there is in a data set, the larger is its standard deviation. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 23

24 Formula 3.1 numerator Σ(x i x) 2 = Σx i 2 (Σx i ) 2 /n Sample variance s 2 = Σ(x i x) 2 n 1 = Σx i 2 (Σx i ) 2 /n n 1 Sample standard deviation s = Σ(x i n 1 x) 2 = Σx i 2 (Σx i ) 2 /n n 1 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 24

25 Tables 3.10 & 3.11 Data sets that have different variation Means and standard deviations of the data sets in Table 3.10 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 25

26 Figures 3.5 and 3.6 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 26

27 Key Fact 3.2 Three-Standard-Deviations Rule Almost all the observations in any data set lie within three standard deviations to either side of the mean. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 27

28 Section 3.3 Chebyshev s Rule and the Empirical Rule Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 28

29 Key Fact 3.3 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 29

30 Key Fact 3.4 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 30

31 Figure 3.9 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 31

32 Table 3.12 PCB concentrations, in parts per million, of 60 pelican eggs Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 32

33 Figure 3.10 Histogram of the PCB-concentration data Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 33

34 Figure 3.10 The mean and one, two, and three standard deviations to either side of the mean for the PCB-concentration data. Recall: x = , s = out of the 60 observations, or 71.7%, lie within one standard deviation to either side of the mean. 57 out of the 60 observations, or 95%, lie within two standard deviations to either side of the mean. 60 out of the 60 observations, or 100%, lie within three standard deviations to either side of the mean. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 34

35 Do Exercise on Page 124 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 35

36 Section 3.4 The Five-Number Summary; Boxplots Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 36

37 Figure 3.12 Quartiles for (a) uniform, (b) bell-shaped, (c) right-skewed, and (d) left-skewed distributions Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 37

38 Definition 3.7 Quartiles First, arrange the data in increasing order. Next, determine the median. Then, divide the (ordered) data set into two halves, a bottom half and a top half; if the number of observations is odd, include the median in both halves. The first quartile (Q 1 ) is the median of the bottom half of the data set. The second quartile (Q 2 ) is the median of the entire data set. The third quartile (Q 3 ) is the median of the top half of the data set. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 38

39 Procedure 3.1 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 39

40 Definition 3.8 Interquartile Range The interquartile range, or IQR, is the difference between the first and third quartiles; that is, IQR = Q 3 Q 1. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 40

41 Definition 3.9 Five-Number Summary The five-number summary of a data set is Min, Q 1, Q 2, Q 3, Max. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 41

42 Definition 3.10 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 42

43 Procedure 3.2 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 43

44 Figure 3.14 Constructing a boxplot for TV viewing times in Table 3.13 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 44

45 Figure 3.15 Boxplots for the data in Table 3.15 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 45

46 Figure 3.16 Boxplots for (a) right-skewed, (b) symmetric, and (d) left-skewed distributions Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 46

47 Section 3.5 Descriptive Measures for Populations; Use of Samples Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 47

48 Definition 3.11 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 48

49 Definition 3.12 Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 49

50 Figure 3.18 Population and sample for bolt diameters Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 50

51 Definition 3.13 Parameter and Statistic Parameter: A descriptive measure for a population. Statistic: A descriptive measure for a sample. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 51

52 Definition 3.14 & 3.15 z-score For an observed value of a variable x, the corresponding value of the standardized variable z is called the z-score of the observation. The term standard score is often used instead of z-score. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 52