1. 1 1 Slide STATISTICS FOR BUSINESS AND ECONOMICS Seventh Edition AndersonSweeneyWilliams Slides Prepared by John Loucks © 1999 ITP/South-Western College Publishing

2. 2 2 Slide Chapter 2 Descriptive Statistics I: Tabular and Graphical Methods n Summarizing Qualitative Data n Summarizing Quantitative Data n Exploratory Data Analysis n Cross-tabulations and Scatter Diagrams

3. 3 3 Slide Summarizing Qualitative Data n Frequency Distribution n Relative Frequency and Percent Frequency Distributions Distributions n Bar Graph n Pie Chart

4. 4 4 Slide Frequency Distribution n A frequency distribution is a tabular summary of a set of data showing the frequency (or number) of items in each of several non-overlapping classes. n The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.

5. 5 5 Slide 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. The ratings provided by a sample of 20 quests are shown below. below. Below AverageAverage Above Average Above Average Above Average Above Average Above Average Below Average Below Average AveragePoor Poor Above Average Excellent Above Average AverageAbove Average Average Above AverageAverage

6. 6 6 Slide n Frequency Distribution Rating Frequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20 Example: Marada Inn

7. 7 7 Slide Relative Frequency and Percent Frequency Distributions n The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class. n A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class. n The percent frequency of a class is the relative frequency multiplied by 100. n A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class.

8. 8 8 Slide Example: Marada Inn n Relative Frequency and Percent Frequency Distributions Relative Percent Relative Percent RatingFrequencyFrequency Poor.1010 Below Average.1515 Average.2525 Above Average.4545 Excellent.05 5 Total 1.00 100

9. 9 9 Slide Bar Graph n A bar graph is a graphical device for depicting qualitative data that have been summarized in a frequency, relative frequency, or percent frequency distribution. n On the horizontal axis we specify the labels used for each of the classes. n A frequency, relative frequency, or percent frequency scale can be used for the vertical axis. n Using a bar of fixed width drawn above each class label, we extend the height appropriately. n The bars are separated to emphasize the fact that each class is a separate category.

10. 10 Slide Example: Marada Inn Bar Graph 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 Poor Below Average Below Average Above Average Above Average Excellent Frequency Rating

11. 11 Slide Pie Chart n The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. n First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. n 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.

12. 12 Slide Example: Marada Inn n Pie Chart Average 25% Average 25% Below Average 15% Below Average 15% Poor 10% Poor 10% Above Average 45% Above Average 45% Exc. 5% Exc. 5% Ratings

13. 13 Slide Summarizing Quantitative Data n Frequency Distribution n Relative Frequency and Percent Frequency Distributions n Dot Plot n Histogram n Cumulative Distribution n Ogive

14. 14 Slide Example: Hudson Auto Repair The manager of Hudson would like to get a better picture of the distribution of costs for engine tune-up parts. A sample of 50 customer invoices has been taken and the costs of parts, rounded to the nearest dollar, are listed below.

15. 15 Slide Frequency Distribution n Guidelines for Selecting Number of Classes Use between 5 and 20 classes. Use between 5 and 20 classes. Data sets with a larger number of elements usually require a larger number of classes. Data sets with a larger number of elements usually require a larger number of classes. Smaller data sets usually require fewer classes. Smaller data sets usually require fewer classes. n Guidelines for Selecting Width of Classes Use classes of equal width. Use classes of equal width. Approximate Class Width = Approximate Class Width =

16. 16 Slide Example: Hudson Auto Repair n Frequency Distribution If we choose six classes, Approximate Class Width = (109 - 52)/6 = 9.5  10 Approximate Class Width = (109 - 52)/6 = 9.5  10 Cost ($)Frequency 50-59 2 50-59 2 60-69 13 60-69 13 70-79 16 70-79 16 80-89 7 80-89 7 90-99 7 90-99 7 100-109 5 Total 50 Total 50

17. 17 Slide Example: Hudson Auto Repair n Relative Frequency Relative Frequency of a Class = Frequency of the Class Frequency of the Classn where: n = total number of elements

18. 18 Slide n Relative Frequency and Percent Frequency Distributions Relative Percent Relative Percent Cost ($) Frequency Frequency Cost ($) Frequency Frequency 50-59.04 4 50-59.04 4 60-69.2626 60-69.2626 70-79.3232 70-79.3232 80-89.1414 80-89.1414 90-99.1414 90-99.1414 100-109.1010 100-109.1010 Total 1.00 100 Total 1.00 100 Example: Hudson Auto Repair

19. 19 Slide Dot Plot n One of the simplest graphical summaries of data is a dot plot. n A horizontal axis shows the range of data values. n Then each data value is represented by a dot placed above the axis.

20. 20 Slide Example: Hudson Auto Repair n Dot Plot 50 60 70 80 90 100 110................................................... Cost ($)

21. 21 Slide Histogram n Another common graphical presentation of quantitative data is a histogram. n The variable of interest is placed on the horizontal axis and the frequency, relative frequency, or percent frequency is placed on the vertical axis. n A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency. n Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes.

22. 22 Slide Example: Hudson Auto Repair Histogram 2 2 4 4 6 6 8 8 10 12 14 16 18 Frequency 50 60 70 80 90 100 110 Cost ($)

23. 23 Slide Cumulative Distribution n The cumulative frequency distribution shows the number of items with values less than or equal to the upper limit of each class. n The cumulative relative frequency distribution shows the proportion of items with values less than or equal to the upper limit of each class. n The cumulative percent frequency distribution shows the percentage of items with values less than or equal to the upper limit of each class.

24. 24 Slide Example: Hudson Auto Repair n Cumulative Distributions Cumulative Cumulative Cumulative Relative Percent Cumulative Relative Percent Cost ($) Frequency Frequency Frequency < 59 2.04 4 < 59 2.04 4 < 69 15.30 30 < 69 15.30 30 < 79 31.62 62 < 79 31.62 62 < 89 38.76 76 < 89 38.76 76 < 99 45.90 90 < 99 45.90 90 < 109 50 1.00 100 < 109 50 1.00 100

25. 25 Slide Ogive n An ogive is a graph of a cumulative frequency or cumulative relative frequency distribution. n The data values are shown on the horizontal axis. n Either the cumulative frequencies or cumulative relative frequencies are shown on the vertical axis.

26. 26 Slide Example: Hudson Auto Repair n Ogive 10 20 30 40 50 Cumulative Frequency 50 60 70 80 90 100 110 Cost ($)

27. 27 Slide Exploratory Data Analysis n The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly.

28. 28 Slide Stem-and-Leaf Display n A stem-and-leaf display is an example of how simple arithmetic and easy-to-draw pictures can be used to summarize data quickly. n This display shows both the rank order and shape of the distribution of the data. n It is similar to a histogram on its side, but it has the advantage of showing the actual data values. n The first digits of each data item are arranged to the left of a vertical line. n To the right of the vertical line we record the last digit for each item in rank order.

29. 29 Slide Example: Hudson Auto Repair n Stem-and-Leaf Display 5 2 7 5 2 7 6 2 2 2 2 5 6 7 8 8 8 9 9 9 6 2 2 2 2 5 6 7 8 8 8 9 9 9 7 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 7 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 8 0 0 2 3 5 8 9 8 0 0 2 3 5 8 9 9 1 3 7 7 7 8 9 9 1 3 7 7 7 8 9 10 1 4 5 5 9 10 1 4 5 5 9

30. 30 Slide Crosstabulations n Crosstabulation is a method that can be used to summarize the data for two variables simultaneously. n Typically, the table’s left and top margin labels define the classes for the two variables. n Crosstabulation can provide insight about the relationship between the variables.

31. 31 Slide Crosstabulations n Crosstabulation of Enrollment by Gender and Degree Level at a University Degree Level Degree Level Gender Undergraduate Graduate Doctorate Total Male 7341 1937 172 9450 Female 8294 1688 119 10101 Total 15635 3625 291 19551

32. 32 Slide Scatter Diagram n Scatter Diagram for Engine Size and Gas Mileage of Eight Automobiles 10 15 20 25 30 In-City Gas Mileage (mpg) 02468 10 Engine Size (number of cylinders)

33. 33 Slide The End of Chapter 2