2. 2-1

3. 2-2 Chapter Two Descriptive Statistics McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

4. 2-3 Descriptive Statistics 2.1 Describing the Shape of a Distribution 2.2 Describing Central Tendency 2.3 Measures of Variation 2.4 Percentiles, Quartiles, and Box-and- Whiskers Displays 2.5 Describing Qualitative Data *2.6Using Scatter Plots to Study the Relationship Between Variables *2.7Misleading Graphs and Charts

5. 2-4 2.1 Stem and Leaf Display: Car Mileage Example 2.1: The Car Mileage Case 1 29 8 5 30 1344 12 30 5666889 21 31 001233444 (11) 31 55566777889 17 32 0001122344 7 32 556788 1 33 3

6. 2-5 Stem and Leaf Display: Payment Times Example 2.2: The Accounts Receivable Case 1 10 0 2 11 0 4 12 00 7 13 000 11 14 0000 18 15 0000000 27 16 000000000 (8) 17 00000000 30 18 000000 24 19 00000 19 20 000 16 21 000 13 22 000 10 23 00 8 24 000 5 25 00 3 26 0 2 27 0 1 28 1 29 0

7. 2-6 Histograms Example 2.4: The Accounts Receivable Case Frequency HistogramRelative Frequency Histogram

8. 2-7 The Normal Curve

9. 2-8 Skewness Right SkewedLeft Skewed Symmetric

10. 2-9 Dot Plots Scores on Exams 1 and 2

11. 2-10 2.2 Population Parameters and Sample Statistics A population parameter is number calculated from all the population measurements that describes some aspect of the population. The population mean, denoted , is a population parameter and is the average of the population measurements. A point estimate is a one-number estimate of the value of a population parameter. A sample statistic is number calculated using sample measurements that describes some aspect of the sample.

12. 2-11 Measures of Central Tendency Mean, σ The average or expected value Median, M d The middle point of the ordered measurements Mode, M o The most frequent value

13. 2-12 The Mean Population X 1, X 2, …, X N  Population Mean Sample x 1, x 2, …, x n Sample Mean

14. 2-13 The Sample Mean The sample mean is defined as and is a point estimate of the population mean, .

15. 2-14 Example: Car Mileage Case Example 2.5:Sample mean for first five car mileages from Table 2.1 30.8, 31.7, 30.1, 31.6, 32.1

16. 2-15 The Median The population or sample median is a value such that 50% of all measurements lie above (or below) it. The median M d is found as follows: 1.If the number of measurements is odd, the median is the middlemost measurement in the ordered values. 2.If the number of measurements is even, the median is the average of the two middlemost measurements in the ordered values.

17. 2-16 Example: Sample Median Example 2.6: Internists’ Salaries (x$1000) 127 132 138 141 144 146 152 154 165 171 177 192 241 Since n = 13 (odd,) then the median is the middlemost or 7th measurement, M d =152

18. 2-17 The Mode The mode, M o of a population or sample of measurements is the measurement that occurs most frequently.

19. 2-18 Example: Sample Mode Example 2.2: The Accounts Receivable Case 1 10 0 2 11 0 4 12 00 7 13 000 11 14 0000 18 15 0000000 27 16 000000000 (8) 17 00000000 30 18 000000 24 19 00000 19 20 000 16 21 000 13 22 000 10 23 00 8 24 000 5 25 00 3 26 0 2 27 0 1 28 1 29 0 The value 16 occurs 9 times therefore: Mo = 16

20. 2-19 Relationships Among Mean, Median and Mode

21. 2-20 2.3 Measures of Variation Range Largest minus the smallest measurement Variance The average of the sum of the squared deviations from the mean Standard Deviation The square root of the variance

22. 2-21 The Range Example: Internists’ Salaries (in thousands of dollars) 127 132 138 141 144 146 152 154 165 171 177 192 241 Range = 241 - 127 = 114 ($114,000) Range = largest measurement - smallest measurement

23. 2-22 The Variance Population X 1, X 2, …, X N Population Variance σ2σ2 Sample x 1, x 2, …, x n Sample Variance s2s2

24. 2-23 The Standard Deviation Population Standard Deviation, s:Sample Standard Deviation, s:

25. 2-24 Example: Population Variance/Standard Deviation Population of annual returns for five junk bond mutual funds: 10.0%, 9.4%, 9.1%, 8.3%, 7.8% m= 10.0+9.4+9.1+8.3+7.8 = 44.6 = 8.92% 5 50 = 1.1664+.2304+.3844+1.2544 = 3.068 =.6136 5 5

26. 2-25 Example: Sample Variance/Standard Deviation s 2 = 2.572  4 = 0.643 Example 2.11: Sample variance and standard deviation for first five car mileages from Table 2.1 30.8, 31.7, 30.1, 31.6, 32.1

27. 2-26 The Empirical Rule for Normal Populations If a population has mean m and standard deviation s and is described by a normal curve, then 68.26% of the population measurements lie within one standard deviation of the mean: [m-s, m+s] 95.44% of the population measurements lie within two standard deviations of the mean: [m-2s, m+2s] 99.73% of the population measurements lie within three standard deviations of the mean: [m-3s, m+3s]

28. 2-27 Example: The Empirical Rule Example 2.13: The Car Mileage Case

29. 2-28 Chebyshev’s Theorem Let m and s be a population’s mean and standard deviation, then for any value k>1, At least 100(1 - 1/k 2 )% of the population measurements lie in the interval: [m-ks, m+ks]

30. 2-29 2.4 Percentiles and Quartiles For a set of measurements arranged in increasing order, the pth percentile is a value such that p percent of the measurements fall at or below the value and (100-p) percent of the measurements fall at or above the value. The first quartile Q 1 is the 25th percentile The second quartile (or median) M d is the 50th percentile The third quartile Q 3 is the 75th percentile. The interquartile range IQR is Q 3 - Q 1

31. 2-30 Example: Quartiles 20 customer satisfaction ratings: 1 3 5 5 7 8 8 8 8 8 8 9 9 9 9 9 10 10 10 10 M d = (8+8)/2 = 8 Q 1 = (7+8)/2 = 7.5 Q 3 = (9+9)/2 = 9 IRQ = Q 3 - Q 1 = 9 - 7.5 = 1.5

32. 2-31 Box and Whiskers Plots

33. 2-32 2.5 Describing Qualitative Data

34. 2-33 Population and Sample Proportions Population X 1, X 2, …, X N p Population Proportion Sample x 1, x 2, …, x n Sample Proportion x i = 1 if characteristic present, 0 if not

35. 2-34 Example: Sample Proportion Example 2.16: Marketing Ethics Case 117 out of 205 marketing researchers disapproved of action taken in a hypothetical scenario X = 117, number of researches who disapprove n = 205, number of researchers surveyed Sample Proportion:

36. 2-35 Bar Chart Percentage of Automobiles Sold by Manufacturer, 1970 versus 1997

37. 2-36 Pie Chart Percentage of Automobiles Sold by Manufacturer,1997

38. 2-37 Pareto Chart Pareto Chart of Labeling Defects

39. 2-38 2.6 Scatter Plots Restaurant Ratings: Mean Preference vs. Mean Taste

40. 2-39 2.7 Misleading Graphs and Charts: Scale Break Mean Salaries at a Major University, 1999 - 2002

41. 2-40 Misleading Graphs and Charts: Horizontal Scale Effects Mean Salary Increases at a Major University, 1999-2002

42. 2-41 Descriptive Statistics 2.1 Describing the Shape of a Distribution 2.2 Describing Central Tendency 2.3 Measures of Variation 2.4 Percentiles, Quartiles, and Box-and- Whiskers Displays 2.5 Describing Qualitative Data *2.6Using Scatter Plots to Study the Relationship Between Variables *2.7Misleading Graphs and Charts Summary: