1. 2-1 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

2. 2-2 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistics for Business and Economics Chapter 2 Methods for Describing Sets of Data

3. 2-3 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Contents 1.Describing Qualitative Data 2.Graphical Methods for Describing Quantitative Data 3.Numerical Measures of Central Tendency 4.Numerical Measures of Variability 5.Using the Mean and Standard Deviation to Describe Data

4. 2-4 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Contents 6.Numerical Measures of Relative Standing 7.Methods for Detecting Outliers: Box Plots and z-scores 8.Graphing Bivariate Relationships 9.The Time Series Plot 10.Distorting the Truth with Descriptive Techniques

5. 2-5 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives 1.Describe data using graphs 2.Describe data using numerical measures

6. 2-6 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.1 Describing Qualitative Data

7. 2-7 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Terms A class is one of the categories into which qualitative data can be classified. The class frequency is the number of observations in the data set falling into a particular class. The class relative frequency is the class frequency divided by the total numbers of observations in the data set. The class percentage is the class relative frequency multiplied by 100.

8. 2-8 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Frequency Distribution Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

9. 2-9 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Frequency Distribution Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

10. 2-10 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary Table 1.Lists categories & number of elements in category 2.Obtained by tallying responses in category 3.May show frequencies (counts), % or both Row Is Category Tally: |||| |||| |||| |||| MajorCount Accounting130 Economics20 Management50 Total200

11. 2-11 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Frequency Distribution Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

12. 2-12 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Bar Graph Vertical Bars for Qualitative Variables Bar Height Shows Frequency or % Zero Point Percent Used Also Equal Bar Widths Frequency

13. 2-13 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Frequency Distribution Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

14. 2-14 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Pie Chart 1.Shows breakdown of total quantity into categories 2.Useful for showing relative differences 3.Angle size (360°)(percent) Econ. 10% Mgmt. 25% Acct. 65% Majors (360°) (10%) = 36° 36°

15. 2-15 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Frequency Distribution Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

16. 2-16 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Pareto Diagram Like a bar graph, but with the categories arranged by height in descending order from left to right. Vertical Bars for Qualitative Variables Bar Height Shows Frequency or % Zero Point Percent Used Also Equal Bar Widths Frequency

17. 2-17 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary Bar graph: The categories (classes) of the qualitative variable are represented by bars, where the height of each bar is either the class frequency, class relative frequency, or class percentage. Pie chart: The categories (classes) of the qualitative variable are represented by slices of a pie (circle). The size of each slice is proportional to the class relative frequency. Pareto diagram: A bar graph with the categories (classes) of the qualitative variable (i.e., the bars) arranged by height in descending order from left to right.

18. 2-18 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re an analyst for IRI. You want to show the market shares held by Web browsers in 2006. Construct a bar graph, pie chart, & Pareto diagram to describe the data. BrowserMkt. Share (%) Firefox14 Internet Explorer81 Safari4 Others1

19. 2-19 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Bar Graph Solution* Market Share (%) Browser

20. 2-20 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Pie Chart Solution* Market Share

21. 2-21 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Pareto Diagram Solution* Market Share (%) Browser

22. 2-22 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.2 Graphical Methods for Describing Quantitative Data

23. 2-23 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

24. 2-24 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Dot Plot 1.Horizontal axis is a scale for the quantitative variable, e.g., percent. 2.The numerical value of each measurement is located on the horizontal scale by a dot.

25. 2-25 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

26. 2-26 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Stem-and-Leaf Display 1. Divide each observation into stem value and leaf value Stems are listed in order in a column Leaf value is placed in corresponding stem row to right of bar 2. Data: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 26 2144677 3028 41

27. 2-27 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Data Presentation Qualitative Data Quantitative Data Summary Table Stem-&-Leaf Display Histogram Bar Graph Pie Chart Pareto Diagram Dot Plot

28. 2-28 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

29. 2-29 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Histogram 0 1 2 3 4 5 Frequency Relative Frequency Percent 015.525.535.545.555.5 Lower Boundary Bars Touch ClassFreq. 15.5 – 25.53 25.5 – 35.55 35.5 – 45.52 Count

30. 2-30 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary Dot plot: The numerical value of each quantitative measurement in the data set is represented by a dot on a horizontal scale. When data values repeat, the dots are placed above one another vertically. Stem-and-leaf display: The numerical value of the quantitative variable is partitioned into a “stem” and a “leaf.” The possible stems are listed in order in a column. The leaf for each quantitative measurement in the data set is placed in the corresponding stem row. Leaves for observations with the same stem value are listed in increasing order horizontally.

31. 2-31 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary Histogram: The possible numerical values of the quantitative variable are partitioned into class intervals, where each interval has the same width. These intervals form the scale of the horizontal axis. The frequency or relative frequency of observations in each class interval is determined. A horizontal bar is placed over each class interval, with height equal to either the class frequency or class relative frequency.

32. 2-32 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.3 Numerical Measures of Central Tendency

33. 2-33 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge... employees cite low pay -- most workers earn only $20,000.... President claims average pay is $70,000! $400,000 $70,000 $50,000 $30,000 $20,000

34. 2-34 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Two Characteristics The central tendency of the set of measurements–that is, the tendency of the data to cluster, or center, about certain numerical values. Central Tendency (Location)

35. 2-35 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Two Characteristics The variability of the set of measurements–that is, the spread of the data. Variation (Dispersion)

36. 2-36 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Standard Notation MeasureSamplePopulation Mean  X  SizenN

37. 2-37 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Mean 1.Most common measure of central tendency 2.Acts as ‘balance point’ 3.Affected by extreme values (‘outliers’) 4.Denoted where x x n xxx n i i n n     1 12 … x

38. 2-38 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Mean Example Raw Data:10.34.98.911.76.37.7 x x n xxxxxx i i n        1 123456 6 10349891176377 6 830.......

39. 2-39 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Median 1.Measure of central tendency 2.Middle value in ordered sequence If n is odd, middle value of sequence If n is even, average of 2 middle values 3.Position of median in sequence 4.Not affected by extreme values Positioning Point  n1 2

40. 2-40 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Median Example Odd-Sized Sample Raw Data:24.122.621.523.722.6 Ordered:21.522.622.623.724.1 Position:12345 Positioning Point Median       n1 2 51 2 30 226..

41. 2-41 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Median Example Even-Sized Sample Raw Data:10.34.98.911.76.37.7 Ordered:4.96.37.78.910.311.7 Position:123456 Positioning Point Median         n1 2 61 2 35 7789 2 830....

42. 2-42 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Mode 1.Measure of central tendency 2.Value that occurs most often 3.Not affected by extreme values 4.May be no mode or several modes 5.May be used for quantitative or qualitative data

43. 2-43 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Mode Example No Mode Raw Data:10.34.98.911.76.37.7 One Mode Raw Data:6.34.98.9 6.3 4.94.9 More Than 1 Mode Raw Data:212828414343

44. 2-44 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. Describe the stock prices in terms of central tendency.

45. 2-45 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Tendency Solution* Mean x x n xxx i i n        1 128 8 1716211813161211 8 155 ….

46. 2-46 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Tendency Solution* Median Raw Data:1716211813161211 Ordered:1112131616171821 Position:12345678 Positioning Point Median         n1 2 81 2 45 16 2.

47. 2-47 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Tendency Solution* Mode Raw Data:1716211813161211 Mode = 16

48. 2-48 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary of Central Tendency Measures MeasureFormulaDescription Mean  x i /n Balance Point Median(n+1) Position 2 Middle Value When Ordered ModenoneMost Frequent

49. 2-49 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Shape 1.Describes how data are distributed 2.Measures of Shape Skew = Symmetry Right-SkewedLeft-SkewedSymmetric Mean =Median Mean Median Median Mean

50. 2-50 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.4 Numerical Measures of Variability

51. 2-51 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Range 1.Measure of dispersion 2.Difference between largest & smallest observations Range = x largest – x smallest 3.Ignores how data are distributed 7891078910 Range = 10 – 7 = 3

52. 2-52 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Variance & Standard Deviation 1.Measures of dispersion 2.Most common measures 3.Consider how data are distributed 461012 x = 8.3 4. Show variation about mean (x or μ) 8

53. 2-53 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Standard Notation MeasureSamplePopulation Mean  x  Standard Deviation s  Variance s 2  2 SizenN

54. 2-54 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Variance Formula n – 1 in denominator!

55. 2-55 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Standard Deviation Formula

56. 2-56 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Variance Example Raw Data:10.34.98.911.76.37.7 s xx n x x n s i i n i i n 2 2 11 2 222 1 83 1038349837783 61 6368           () ()()() where........ …

57. 2-57 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the variance and standard deviation of the stock prices?

58. 2-58 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Variation Solution* Sample Variance Raw Data:1716211813161211 s xx n x x n s i i n i i n 2 2 11 2 22 2 1 155 171551615511155 81 1114           () ( ) ( ) ( ) where..... …

59. 2-59 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Variation Solution* Sample Standard Deviation

60. 2-60 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary of Variation Measures MeasureFormulaDescription Range X largest –X smallest Total Spread Standard Deviation (Sample) Dispersion about Sample Mean Standard Deviation (Population) Dispersion about Population Mean Variance (Sample) Squared Dispersion about Sample Mean

61. 2-61 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.5 Using the Mean and Standard Deviation to Describe Data

62. 2-62 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interpreting Standard Deviation: Chebyshev’s Theorem Applies to any shape data set No useful information about the fraction of data in the interval x – s to x + s At least 3/4 of the data lies in the interval x – 2s to x + 2s At least 8/9 of the data lies in the interval x – 3s to x + 3s In general, for k > 1, at least 1 – 1/k 2 of the data lies in the interval x – ks to x + ks

63. 2-63 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interpreting Standard Deviation: Chebyshev’s Theorem No useful information At least 3/4 of the dataAt least 8/9 of the data

64. 2-64 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Chebyshev’s Theorem Example Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is 3.34. Use this information to form an interval that will contain at least 75% of the closing stock prices of new stock issues.

65. 2-65 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Chebyshev’s Theorem Example At least 75% of the closing stock prices of new stock issues will lie within 2 standard deviations of the mean. x = 15.5 s = 3.34 (x – 2s, x + 2s) = (15.5 – 2∙3.34, 15.5 + 2∙3.34) = (8.82, 22.18)

66. 2-66 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interpreting Standard Deviation: Empirical Rule Applies to data sets that are mound shaped and symmetric Approximately 68% of the measurements lie in the interval Approximately 95% of the measurements lie in the interval Approximately 99.7% of the measurements lie in the interval

67. 2-67 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interpreting Standard Deviation: Empirical Rule x – 3s x – 2s x – s x x + s x +2s x + 3s Approximately 68% of the measurementsApproximately 95% of the measurements Approximately 99.7% of the measurements

68. 2-68 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Empirical Rule Example Previously we found the mean closing stock price of new stock issues is 15.5 and the standard deviation is 3.34. If we can assume the data is symmetric and mound shaped, calculate the percentage of the data that lie within the intervals x + s, x + 2s, x + 3s.

69. 2-69 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Empirical Rule Example Approximately 95% of the data will lie in the interval (x – 2s, x + 2s), (15.5 – 2∙3.34, 15.5 + 2∙3.34) = (8.82, 22.18) Approximately 99.7% of the data will lie in the interval (x – 3s, x + 3s), (15.5 – 3∙3.34, 15.5 + 3∙3.34) = (5.48, 25.52) According to the Empirical Rule, approximately 68% of the data will lie in the interval (x – s, x + s), (15.5 – 3.34, 15.5 + 3.34) = (12.16, 18.84)

70. 2-70 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.6 Numerical Measures of Relative Standing

71. 2-71 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Numerical Measures of Relative Standing: Percentiles Describes the relative location of a measurement compared to the rest of the data The p th percentile is a number such that p% of the data falls below it and (100 – p)% falls above it Median = 50 th percentile

72. 2-72 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Percentile Example You scored 560 on the GMAT exam. This score puts you in the 58 th percentile. What percentage of test takers scored lower than you did? What percentage of test takers scored higher than you did?

73. 2-73 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Percentile Example What percentage of test takers scored lower than you did? 58% of test takers scored lower than 560. What percentage of test takers scored higher than you did? (100 – 58)% = 42% of test takers scored higher than 560.

74. 2-74 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Numerical Measures of Relative Standing: z–Scores Describes the relative location of a measurement compared to the rest of the data Measures the number of standard deviations away from the mean a data value is located Sample z–scorePopulation z–score

75. 2-75 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. z–Score Example The mean time to assemble a product is 22.5 minutes with a standard deviation of 2.5 minutes. Find the z–score for an item that took 20 minutes to assemble. Find the z–score for an item that took 27.5 minutes to assemble.

76. 2-76 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. z–Score Example x = 20, μ = 22.5 σ = 2.5 x – μ 20 – 22.5 σ z = = 2.5 = –1.0 x = 27.5, μ = 22.5 σ = 2.5 x – μ 27.5 – 22.5 σ z = = 2.5 = 2.0

77. 2-77 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interpretation of z–Scores for Mound-Shaped Distributions of Data 1.Approximately 68% of the measurements will have a z-score between –1 and 1. 2.Approximately 95% of the measurements will have a z-score between –2 and 2. 3.Approximately 99.7% of the measurements will have a z-score between –3 and 3. (see the figure on the next slide)

78. 2-78 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interpretation of z–Scores

79. 2-79 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.7 Methods for Detecting Outliers: Box Plots and z-Scores

80. 2-80 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Outlier An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier. Outliers typically are attributable to one of the following causes: 1.The measurement is observed, recorded, or entered into the computer incorrectly. 2.The measurement comes from a different population. 3.The measurement is correct but represents a rare (chance) event.

81. 2-81 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Quartiles Measure of noncentral tendency 25%25%25%25% Q1Q1Q1Q1 Q2Q2Q2Q2 Q3Q3Q3Q3 Split ordered data into 4 quarters Lower quartile Q L is 25 th percentile. Middle quartile m is the median. Upper quartile Q U is 75 th percentile. Interquartile range: IQR = Q U – Q L

82. 2-82 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Quartile (Q 2 ) Example Raw Data:10.34.98.911.76.37.7 Ordered:4.96.37.78.910.311.7 Position:123456 Q 2 is the median, the average of the two middle scores (7.7 + 8.9)/2 = 8.3

83. 2-83 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Quartile (Q 1 ) Example Raw Data:10.34.98.911.76.37.7 Ordered:4.96.37.78.910.311.7 Position:123456 Q L is median of bottom half = 6.3

84. 2-84 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Quartile (Q 3 ) Example Raw Data:10.34.98.911.76.37.7 Ordered:4.96.37.78.910.311.7 Position:123456 Q U is median of bottom half = 10.3

85. 2-85 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interquartile Range 1.Measure of dispersion 2.Also called midspread 3.Difference between third & first quartiles Interquartile Range = Q 3 – Q 1 4.Spread in middle 50% 5.Not affected by extreme values

86. 2-86 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re a financial analyst for Prudential-Bache Securities. You have collected the following closing stock prices of new stock issues: 17, 16, 21, 18, 13, 16, 12, 11. What are the quartiles, Q 1 and Q 3, and the interquartile range?

87. 2-87 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Quartile Solution* Q 1 Raw Data:1716211813161211 Ordered:1112131616171821 Position:12345678 Q L is the median of the bottom half, the average of the two middle scores (12 + 13)/2 = 12.5

88. 2-88 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Quartile Solution* Q 3 Raw Data:1716211813161211 Ordered:1112131616171821 Position:12345678 Q U is the median of the bottom half, the average of the two middle scores (17 + 18)/2 = 17.5

89. 2-89 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Interquartile Range Solution* Interquartile Range Raw Data:1716211813161211 Ordered:1112131616171821 Position:12345678 Interquartile Range = Q 3 – Q 1 = 17.5 – 12.5 = 5

90. 2-90 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Box Plot 1.Graphical display of data using 5-number summary Median 4681012 Q 3 Q 1 X largest X smallest

91. 2-91 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Box Plot 1.Draw a rectangle (box) with the ends (hinges) drawn at the lower and upper quartiles (Q L and Q U ). The median data is shown by a line or symbol (such as “+”). 2.The points at distances 1.5(IQR) from each hinge define the inner fences of the data set. Line (whiskers) are drawn from each hinge to the most extreme measurements inside the inner fence.

92. 2-92 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Box Plot 3.A second pair of fences, the outer fences, are defined at a distance of 3(IQR) from the hinges. One symbol (*) represents measurements falling between the inner and outer fences, and another (0) represents measurements beyond the outer fences. 4.Symbols that represent the median and extreme data points vary depending on software used. You may use your own symbols if you are constructing a box plot by hand.

93. 2-93 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Shape & Box Plot Right-SkewedLeft-SkewedSymmetric Q 1 Median Q 3 Q 1 Median Q 3 Q 1 Median Q 3

94. 2-94 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Detecting Outliers Box Plots: Observations falling between the inner and outer fences are deemed suspect outliers. Observations falling beyond the outer fence are deemed highly suspect outliers. z-scores: Observations with z-scores greater than 3 in absolute value are considered outliers. (For some highly skewed data sets, observations with z-scores greater than 2 in absolute value may be outliers.)

95. 2-95 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 2.10 Distorting the Truth with Descriptive Statistics

96. 2-96 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Errors in Presenting Data 1.Use area to equate to value 2.No relative basis in comparing data batches 3.Compress the vertical axis 4.No zero point on the vertical axis 5.Gap in the vertical axis 6.Use of misleading wording 7.Knowing central tendency without knowing variability

97. 2-97 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Reader Equates Area to Value Bad Presentation Good Presentation 1960: $1.00 1970: $1.60 1980: $3.10 1990: $3.80 Minimum Wage 0 2 4 1960197019801990 $

98. 2-98 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. No Relative Basis Good Presentation A’s by Class Bad Presentation 0 100 200 300 FRSOJRSR Freq. 0% 10% 20% 30% FRSOJRSR %

99. 2-99 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Compressing Vertical Axis Good Presentation Quarterly Sales Bad Presentation 0 25 50 Q1Q2Q3Q4 $ 0 100 200 Q1Q2Q3Q4 $

100. 2-100 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. No Zero Point on Vertical Axis Good Presentation Monthly Sales Bad Presentation 0 20 40 60 JMMJSN $ 36 39 42 45 JMMJSN $

101. 2-101 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Gap in the Vertical Axis Bad Presentation

102. 2-102 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Changing the Wording Changing the title of the graph can influence the reader. We’re not doing so well.Still in prime years!

103. 2-103 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Knowing only central tendency Knowing ONLY the central tendency might lead one to purchase Model A. Knowing the variability as well may change one’s decision!

104. 2-104 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Describing Qualitative Data 1. Identify category classes 2.Determine class frequencies 3.Class relative frequency = (class freq)/n 4.Graph relative frequencies

105. 2-105 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Graphing Quantitative Data 1 Variable 1. Identify class intervals 2.Determine class interval frequencies 3.Class relative relative frequency = (class interval frequencies)/n 4.Graph class interval relative frequencies

106. 2-106 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Graphing Quantitative Data 2 Variables Scatterplot

107. 2-107 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Numerical Description of Quantitative Data Central Tendency Mean Median Mode

108. 2-108 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Numerical Description of Quantitative Data Variation Range Variance Standard Deviation Interquartile range

109. 2-109 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Numerical Description of Quantitative Data Relative standing Percentile score z-score

110. 2-110 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Rules for Detecting Quantitative Outliers Interval Chebyshev’s Rule Empirical Rule At least 0% At least 75% At least 89% ≈ 68% ≈ 95% All

111. 2-111 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Rules for Detecting Quantitative Outliers Method Suspect Highly Suspect Values between inner and outer fences 2 < |z| < 3 Box plot: z-score Values beyond outer fences |z| > 3