Statistics

1. Statistics Descriptive Statistics—Tables and Graphics

2. Contents • Summarizing Qualitative Data • Summarizing Quantitative Data • The Stem-and-Leaf Display(枝葉圖) • Cross-tabulations and Scatter Diagrams

3. STATISTICS in PRACTICE • The Colgate-Palmolive Company uses statistics in its quality assurance program for home laundry detergent products. • One concern is customer satisfaction with the quantity of detergent in a carton. To control the problem of heavy detergent powder, limits are placed on the acceptable range of powder density.

4. STATISTICS in PRACTICE • Statistical samples are taken and the density of each powder sample is measured. • Data summaries are then provided for operating personnel so that corrective action can be taken if necessary to keep the density within the desired quality.

5. Summarizing Qualitative Data • Frequency Distribution • Relative Frequency Distributions • Percent Frequency Distributions • Bar Graphs • Pie Charts

6. Summarizing Qualitative Data -- Frequency Distribution A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several nonoverlapping classes. The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.

7. Summarizing Qualitative Data -- Frequency Distribution • Example: Data from a sample of 50 Soft Drink Purchases

8. Summarizing Qualitative Data -- Frequency Distribution • Frequency Distribution Soft Drink Frequency Coke Classic 19 Diet Coke 8 Dr. Pepper 5 Pepsi-Cola 13 Sprite 5 Total 50

9. Summarizing Qualitative Data -- Frequency Distribution • Example: Marada Inn, Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor.

10. Summarizing Qualitative Data -- Frequency Distribution • The ratings provided by a sample of 20 guests are: Above Average Above Average Below Average Poor Above Average Average Average Average Above Average Below Average Poor Excellent Above Average Average Below Average Above Average Above Average Average Above Average Average Above Average

11. Summarizing Qualitative Data -- Frequency Distribution Poor Below Average Average Above Average Excellent Rating Frequency 2 3 5 9 1 Total 20

12. Frequency of the class Relative frequency of a class= n Summarizing Qualitative Data – Relative Frequency The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class.

13. Summarizing Qualitative Data – Relative Frequency Distribution • A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

14. Summarizing Qualitative Data – Relative Frequency Distribution Example: Relative and Percent Frequency Distribution of Soft Drink Purchases Soft Drink Relative Frequency Coke Classic .38 Diet Coke .16 Dr. Pepper .10 Pepsi-Cola .26 Sprite .10 Total 1.00

15. Summarizing Qualitative Data – Percent Frequency Distribution • The percent frequency of a class is the • relative frequency multiplied by 100. • Apercent frequency distribution is a tabular • summary of a set of data showing the percent • frequency for each class.

16. Summarizing Qualitative Data – Percent Frequency Distribution Example: Percent Frequency Distribution of Soft Drink Purchases Soft Drink Percent Frequency Coke Classic 38 Diet Coke 16 Dr. Pepper 10 Pepsi-Cola 26 Sprite 10 Total 100

17. Summarizing Qualitative Data – Relative and Percent Frequency Distribution Relative Frequency Percent Frequency Rating Poor Below Average Average Above Average Excellent 10 15 25 45 5 100 .10 .15 .25 .45 .05 Total 1.00 .10(100) = 10 1/20 = .05

18. Summarizing Qualitative Data – Bar Graph • A bar graph is a graphical device for depicting qualitative data. • On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes. • A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the vertical axis) • Using a bar of fixed width drawn above each class label, we extend the height appropriately.

19. Summarizing Qualitative Data – Bar Graph • Example: Bar Graph of Soft Drink Purchases

20. 10 9 8 7 6 5 4 3 2 1 Summarizing Qualitative Data – Bar Graph Marada Inn Quality Ratings Frequency Rating Excellent Poor Average Above Average Below Average

21. Summarizing Qualitative Data – Pie Chart • The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. • First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. • Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle.

22. Summarizing Qualitative Data – Pie Chart • Example: Pie Chart of Soft Drink Purchases

23. Summarizing Qualitative Data – Pie Chart--Example Marada InnQuality Ratings Excellent 5% Poor 10% Below Average 15% Above Average 45% Average 25%

24. Summarizing Qualitative Data – Pie Chart--Example • Insights Gained from the Preceding Pie Chart • One-half of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager. • For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager.

25. Summarizing Quantitative Data • Frequency Distribution • Relative Frequency and Percent Frequency Distributions • Dot Plot • Histogram • Cumulative Distributions • Ogive(折線圖)

26. Summarizing Quantitative Data-- Frequency Distribution • The three steps necessary to define the classes for a frequency distribution with quantitative data are: 1. Determine the number of nonoverlapping classes. we recommend using between 5 and 20 classes. 2. Determine the width of each class. 3. Determine the class limits. Largest data value –Smallest data value Approximate class width= Number of classes

27. Summarizing Quantitative Data-- Frequency Distribution • Use between 5 and 20 classes. • Data sets with a larger number of elements usually require a larger number of classes. • Smaller data sets usually require fewer classes

28. Summarizing Quantitative Data-- Frequency Distribution • Guidelines for Selecting Width of Classes Use classes of equal width. Approximate Class Width = Largest data value –Smallest data value Number of classes

29. YEAR-END AUDIT TIMES (IN DAYS) 12 14 19 18 15 15 18 17 20 27 22 23 22 21 33 28 14 18 16 13 • Summarizing Quantitative Data-- Frequency Distribution • Example: These data show the time in days required to complete year-end audits for a sample of 20 clients of Sanderson and Clifford, a small public accounting firm with the data rounded to the nearest day.

30. Audit Time Frequency (days) 10-14 4 15-19 8 20-24 5 25-29 2 30-34 1 Total 20 • Summarizing Quantitative Data-- Frequency Distribution • Example: 1. Number of classes = 5 2. provides an approximate class width of (33 — 12)/5= 4.2. 3. We therefore decided to round up and use a class width of five days in the frequency distribution.

31. Summarizing Quantitative Data-- Frequency Distribution • The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide.

32. Summarizing Quantitative Data-- Frequency Distribution • Sample of Parts Cost for 50 Tune-ups

33. Summarizing Quantitative Data-- Frequency Distribution For Hudson Auto Repair, if we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5  10 2 13 16 7 7 5 Total 50 50-59 60-69 70-79 80-89 90-99 100-109 Parts Cost ($) Frequency

34. Summarizing Quantitative Data-- Relative Frequency andPercent Frequency Distributions Relative Frequency Percent Frequency Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 .04 .26 .32 .14 .14 .10 Total 1.00 4 26 32 14 14 10 100 .04(100) 2/50

35. Summarizing Quantitative Data-- Relative Frequency andPercent Frequency Distributions • Insights Gained from the Percent Frequency • Distribution • Only 4% of the parts costs are in the $50-59 class. • 30%of the parts costs are under $70. • The greatest percentage (32% or almost one-third) • of the parts costs are in the $70-79 class. • 10% of the parts costs are $100 or more.

36. Dot Plot • One of the simplest graphical summaries of data is a dot plot. • A horizontal axis shows the range of data values. • Then each data value is represented by a dot placed above the axis. • Example: Dot Plot for The Audit Time Data

37. Dot Plot

38. . . .. . . . . .. .. .. .. . . . . . ..... .......... .. . .. . . ... . .. . Dot Plot Tune-up Parts Cost 50 60 70 80 90 100 110 Cost ($)

39. Histogram • Another common graphical presentation of • quantitative data is a histogram. • The variable of interest is placed on the • horizontalaxis. • A rectangle is drawn above each class interval • withits height corresponding to the interval’s • frequency , relative frequency, or percent • frequency. • Unlike a bar graph, a histogram has no natural • separation between rectangles of adjacent classes.

40. Histogram • Example: Histogram for The Audit Time Data

41. 18 16 14 12 10 8 6 4 2 Histogram Tune-up Parts Cost Frequency Parts Cost ($) 50-59 60-69 70-79 80-89 90-99 100-110

42. .35 .30 .25 .20 .15 .10 .05 0 Histogram • Histogram provides information about the shape. • Symmetric • Left tail is the mirror image of the right tail • Examples: heights and weights of people Relative Frequency

43. .35 .30 .25 .20 .15 .10 .05 0 Histogram • Moderately Skewed Left • A longer tail to the left • Example: exam scores Relative Frequency

44. .35 .30 .25 .20 .15 .10 .05 0 Histogram • Moderately Right Skewed • A Longer tail to the right • Example: housing values Relative Frequency

45. .35 .30 .25 .20 .15 .10 .05 0 Histogram • Highly Skewed Right • A very long tail to the right • Example: executive salaries Relative Frequency

46. Cumulative Distributions • Cumulative frequency distribution - shows the • number of items with values less than or equal to • the upper limit of each class.. • Cumulative relative frequency distribution – • showsthe proportion of items with values less • than or equal to the upper limit of each class. • Cumulative percent frequency distribution – • showsthe percentage of items with values less • than or equal to the upper limit of each class.

47. Cumulative Distributions • Example: Cumulative Frequency, Cumulative Relative Frequency and Cumulative Percent Frequency Distributions for the Audit Data.

48. Cumulative Distributions • Hudson Auto Repair Cumulative Relative Frequency Cumulative Percent Frequency Cumulative Frequency < 59 < 69 < 79 < 89 < 99 < 109 Cost ($) 2 15 31 38 45 50 .04 .30 .62 .76 .90 1.00 4 30 62 76 90 100 2 + 13 15/50 .30(100)

49. Ogive • An ogive is a graph of a cumulative distribution. • The data values are shown on the horizontal axis. • Shown on the vertical axis are the: • cumulative frequencies, or • cumulative relative frequencies, or • cumulative percent frequencies

50. Ogive • The frequency (one of the above) of each class is plotted as a point. • The plotted points are connected by straight lines.