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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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  1. Slides Prepared by JOHN S. LOUCKS St. Edward’s University

  2. Chapter 2Descriptive Statistics:Tabular and Graphical Methods • Summarizing Qualitative Data • Summarizing Quantitative Data • Exploratory Data Analysis • Crosstabulations and Scatter Diagrams

  3. Summarizing Qualitative Data • Frequency Distribution • Relative Frequency • Percent Frequency Distribution • Bar Graphs and Pie Charts

  4. 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.

  5. 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 Average Average Above Average Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor Above Average Excellent Above Average Average Above Average Average Above Average Average

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

  7. Using Excel’s COUNTIF Functionto Construct a Frequency Distribution • Formula Worksheet Note: Rows 9-21 are not shown.

  8. Using Excel’s COUNTIF Functionto Construct a Frequency Distribution • Value Worksheet Note: Rows 9-21 are not shown.

  9. Relative Frequency Distribution • The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class. • A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

  10. 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.

  11. Example: Marada Inn • Relative Frequency and Percent Frequency Distributions RelativePercent RatingFrequencyFrequency Poor .10 10 Below Average .15 15 Average .25 25 Above Average .45 45 Excellent .05 5 Total 1.00 100

  12. Using Excel to Construct Relative Frequency and Percent Frequency Distributions • Formula Worksheet Note: Columns A-B and rows 9-21 and are not shown.

  13. Using Excel to Construct Relative Frequency and Percent Frequency Distributions • Value Worksheet Note: Columns A-B and rows 9-21 and are not shown.

  14. Bar Graph • A bar graph is a graphical device for depicting qualitative data that have been summarized in a frequency, relative frequency, or percent frequency distribution. • On 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 vertical axis. • Using a bar of fixed width drawn above each class label, we extend the height appropriately. • The bars are separated to emphasize the fact that each class is a separate category.

  15. 9 8 7 6 Frequency 5 4 3 2 1 Rating Above Average Excellent Poor Below Average Average Example: Marada Inn • Bar Graph

  16. Using Excel’s Chart Wizardto Construct Bar Graphs Step 1 Select cells C1:D6 Step 2 Select the Chart Wizard button Step 3 When the Chart Type dialog box appears: Choose Column in the Chart type list Choose Clustered Column from the Chart sub-type display Select Next > Step 4 When the Chart Source Data dialog box appears Select Next > … continued

  17. Using Excel’s Chart Wizardto Construct Bar Graphs Step 5 When the Chart Options dialog box appears: Select the Titles tab and then Type Customers’ Quality Ratings in the Chart title box Enter QualityRating in the Value (X) axis box Enter Frequency in the Value (Y) axis box Select the Legend tab and then Remove the check in the Show Legend box Select Next > … continued

  18. Using Excel’s Chart Wizardto Construct Bar Graphs Step 6 When the Chart Location dialog box appears: Specify the location for the new chart Select Finish to display the bar graph

  19. Using Excel’s Chart Wizardto Construct Bar Graphs

  20. 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.

  21. Exc. 5% Poor 10% Below Average 15% Above Average 45% Average 25% Quality Ratings Example: Marada Inn • Pie Chart

  22. Using Excel’s Chart Wizardto Construct Pie Charts Step 1 Select cells C2:C6 and F2:F6 Step 2 Select the Chart Wizard button Step 3 When the Chart Type dialog box appears: Choose Pie in the Chart type list Choose Pie from the Chart sub-type display Select Next > Step 4 When the Chart Source Data dialog box appears Select Next > … continued

  23. Using Excel’s Chart Wizardto Construct Pie Charts Step 5 When the Chart Options dialog box appears: Select the Titles tab and then Type Customers’ Quality Ratingsat Marada in the Chart title box Select the Legend tab and then Remove the check in the Show Legend box Select the Data Labels tab and then Select Show Label and percent Select Show leader lines Select Next > … continued

  24. Using Excel’s Chart Wizardto Construct Pie Charts Step 6 When the Chart Location dialog box appears: Specify the location for the new chart Select Finish to display the pie chart

  25. Using Excel’s Chart Wizardto Construct Pie Charts

  26. Example: Marada Inn • 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.

  27. Summarizing Quantitative Data • Frequency Distribution • Relative Frequency and Percent Frequency Distributions • Dot Plot • Histogram • Cumulative Distributions • Ogive

  28. Example: Hudson Auto Repair The manager of Hudson Auto 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.

  29. Frequency Distribution • Guidelines for Selecting Number of Classes • 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.

  30. Frequency Distribution • Guidelines for Selecting Width of Classes • Use classes of equal width. • Approximate Class Width =

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

  32. Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution • Formula Worksheet (showing data entered) Note: Rows 9-51 are not shown.

  33. Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution • The FREQUENCY function is not a “simple” Excel function. • FREQUENCY is capable of providing multiple values. • In Excel, a formula that can return multiple values is called an array formula. • An array formula must be entered in a special way.

  34. Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution • Entering the Necessary Array Formula Step 1 Select D2:D7 (where the frequencies will appear) Step 2 Type the following formula: =FREQUENCY(A2:A51,{59,69,79,89,99,109}) Step 3 Hold down CTRL and SHIFT keys while pressing ENTER key (Array formula will be entered in D2:D7)

  35. Using Excel’s FREQUENCY Functionto Construct a Frequency Distribution • Value Worksheet Note: Rows 9-51 are not shown.

  36. Example: Hudson Auto Repair • Relative Frequency and Percent Frequency Distributions Relative Percent Cost ($)FrequencyFrequency 50-59 .04 4 60-69 .26 26 70-79 .32 32 80-89 .14 14 90-99 .14 14 100-109 .1010 Total 1.00 100

  37. Example: Hudson Auto Repair • 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.

  38. Histogram • Another common graphical presentation of quantitative data is a histogram. • The variable of interest is placed on the horizontal axis and the frequency, relative frequency, or percent frequency is placed on the vertical axis. • A rectangle is drawn above each class interval with its 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.

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

  40. Using Excel’s Chart Wizardto Construct a Histogram Step 1 Select cells C1:D7 Step 2 Select the Chart Wizard button Step 3 When the Chart Type dialog box appears: Choose Column in the Chart type list Choose Clustered Column from the Chart sub-type display Select Next > Step 4 When the Chart Source Data dialog box appears Select Next > … continued

  41. Using Excel’s Chart Wizardto Construct a Histogram Step 5 When the Chart Options dialog box appears: Select the Titles tab and then Type Histogram for Parts Cost Data in the Chart title box Enter Parts Cost ($) in the Value (X) axis box Enter Frequency in the Value (Y) axis box Select the Legend tab and then Remove the check in the Show Legend box Select Next > … continued

  42. Using Excel’s Chart Wizardto Construct a Histogram Step 6 When the Chart Location dialog box appears: Specify the location for the new chart Select Finish to display the histogram

  43. Using Excel’s Chart Wizardto Construct a Histogram

  44. Using Excel’s Chart Wizardto Construct a Histogram • Eliminating Gaps Between Rectangles Step 1 Right click on any rectangle in the column chart Step 2 Select the Format Data Series option Step 3 When the Format Data Series Option dialog box appears: Select the Options tab and then Enter 0 in the Gap width box Click OK

  45. Using Excel’s Chart Wizardto Construct a Histogram

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

  47. Example: Hudson Auto Repair • Cumulative Distributions Cumulative Cumulative Cumulative Relative Percent Cost ($)FrequencyFrequencyFrequency < 59 2 .04 4 < 69 15 .30 30 < 79 31 .62 62 < 89 38 .76 76 < 99 45 .90 90 < 109 50 1.00 100

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

  49. Example: Hudson Auto Repair • Ogive • Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on. • These gaps are eliminated by plotting points halfway between the class limits. • Thus, 59.5 is used for the 50-59 class, 69.5 is used for the 60-69 class, and so on.

  50. Example: Hudson Auto Repair • Ogive with Cumulative Percent Frequencies 100 80 60 Cumulative Percent Frequency 40 20 Parts Cost ($) 50 60 70 80 90 100 110