Kxu stat-anderson-ch02
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This document discusses various methods for summarizing and exploring qualitative and quantitative data through tabular and graphical techniques, including frequency distributions, relative frequency distributions, bar graphs, pie charts, histograms, scatter plots, and cross-tabulations. It provides examples and explanations of how to construct and interpret these summaries and graphs using sample customer satisfaction and automobile repair data. The goal is to gain insights about relationships within the data that are not evident from just looking at the original values.
Kxu stat-anderson-ch02
- 1. Chapter 2 Descriptive Statistics: Tabular and Graphical Methods Summarizing Qualitative Data Summarizing Quantitative Data Exploratory Data Analysis Crosstabulations and Scatter Diagrams Slide 1
- 2. Summarizing Qualitative Data Frequency Distribution Relative Frequency Percent Frequency Distribution Bar Graph Pie Chart Slide 2
- 3. 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. Slide 3
- 4. 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 Below Average Average Poor Above Average Excellent Average Above Average Above Average Average Above Average Above Average Below Average Poor Above Average Average Slide 4
- 5. Example: Marada Inn Frequency Distribution Rating Frequency Poor 2 Below Average 3 Average 5 Above Average 9 Excellent 1 Total 20 Slide 5
- 6. 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. Slide 6
- 7. Percent Frequency Distribution The percent frequency of a class is the relative frequency multiplied by 100. A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class. Slide 7
- 8. Example: Marada Inn Relative Frequency and Percent Frequency Distributions Rating Poor Below Average Average Above Average Excellent Total Relative Percent Frequency Frequency .10 .15 .25 .45 .05 1.00 10 15 25 45 5 100 Slide 8
- 9. Bar Graph A bar graph is a graphical device for depicting qualitative data. 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. Slide 9
- 10. Example: Marada Inn Bar Graph 9 Frequency 8 7 6 5 4 3 2 1 Poor Below Average Above Excellent Average Average Rating Slide 10
- 11. 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. Slide 11
- 12. Example: Marada Inn Pie Chart Exc. Poor 5% 10% Above Average 45% Below Average 15% Average 25% Quality Ratings Slide 12
- 13. 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. Slide 13
- 14. Summarizing Quantitative Data Frequency Distribution Relative Frequency and Percent Frequency Distributions Dot Plot Histogram Cumulative Distributions Ogive Slide 14
- 15. 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. 91 71 104 85 62 78 69 74 97 82 93 72 62 88 98 57 89 68 68 101 75 66 97 83 79 52 75 105 68 105 99 79 77 71 79 80 75 65 69 69 97 72 80 67 62 62 76 109 74 73 Slide 15
- 16. 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. Slide 16
- 17. 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 Slide 17
- 18. Example: Hudson Auto Repair Frequency Distribution If we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 ≅ 10 Cost ($) 50-59 60-69 70-79 80-89 90-99 100-109 Frequency 2 13 16 7 7 5 Total 50 Slide 18
- 19. Example: Hudson Auto Repair Relative Frequency and Percent Frequency Distributions Relative Cost ($) Frequency 50-59 .04 60-69 .26 70-79 .32 80-89 .14 90-99 .14 100-109 .10 Total 1.00 Percent Frequency 4 26 32 14 14 10 100 Slide 19
- 20. 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 $100 or more. Slide 20
- 21. Dot Plot One of the simplest graphical summaries of data is a dot plot. A horizontal axis shows the range of data values. Then each data value is represented by a dot placed above the axis. Slide 21
- 22. Example: Hudson Auto Repair Dot Plot . 50 . . .. .. .. . . .. . . ..... .......... .. . .. . . ... . .. . .. . . . . 60 70 80 90 100 110 Cost ($) Slide 22
- 23. Histogram Another common graphical presentation of quantitative data is a histogram. The variable of interest is placed on the horizontal 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. Slide 23
- 24. Example: Hudson Auto Repair Histogram 18 Frequency 16 14 12 10 8 6 4 2 50 60 70 80 90 100 110 Parts Cost ($) Slide 24
- 25. Cumulative Distributions Cumulative frequency distribution -- shows the number of items with values less than or equal to the upper limit of each class. Cumulative relative frequency distribution -- shows the proportion of items with values less than or equal to the upper limit of each class. Cumulative percent frequency distribution -- shows the percentage of items with values less than or equal to the upper limit of each class. Slide 25
- 26. Example: Hudson Auto Repair Cumulative Distributions Cost ($) < 59 < 69 < 79 < 89 < 99 < 109 Cumulative Cumulative Cumulative Relative Percent Frequency Frequency Frequency 2 .04 4 15 .30 30 31 .62 62 38 .76 76 45 .90 90 50 1.00 100 Slide 26
- 27. 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. Slide 27
- 28. 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. Slide 28
- 29. Example: Hudson Auto Repair Cumulative Percent Frequency Ogive with Cumulative Percent Frequencies 100 80 60 40 20 50 60 70 80 90 100 110 Parts Cost ($) Slide 29
- 30. Exploratory Data Analysis The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly. One such technique is the stem-and-leaf display. Slide 30
- 31. Stem-and-Leaf Display A stem-and-leaf display shows both the rank order and shape of the distribution of the data. It is similar to a histogram on its side, but it has the advantage of showing the actual data values. The first digits of each data item are arranged to the left of a vertical line. To the right of the vertical line we record the last digit for each item in rank order. Each line in the display is referred to as a stem. Each digit on a stem is a leaf. 8 57 9 3678 Slide 31
- 32. Stem-and-Leaf Display Leaf Units • A single digit is used to define each leaf. • In the preceding example, the leaf unit was 1. • Leaf units may be 100, 10, 1, 0.1, and so on. • Where the leaf unit is not shown, it is assumed to equal 1. Slide 32
- 33. Example: Leaf Unit = 0.1 If we have data with values such as 8.6 11.7 9.4 9.1 10.2 11.0 8.8 a stem-and-leaf display of these data will be Leaf Unit = 0.1 8 6 8 9 1 4 10 2 11 0 7 Slide 33
- 34. Example: Leaf Unit = 10 If we have data with values such as 1806 1717 1974 1791 1682 1910 1838 a stem-and-leaf display of these data will be Leaf Unit = 10 16 8 17 1 9 18 0 3 19 1 7 Slide 34
- 35. Example: Hudson Auto Repair Stem-and-Leaf Display 5 6 7 8 9 10 2 2 1 0 1 1 7 2 1 0 3 4 2 2 2 7 5 2 2 3 7 5 5 3 5 7 9 6 4 8 8 7 8 8 8 9 9 9 4 5 5 5 6 7 8 9 9 9 9 9 Slide 35
- 36. Stretched Stem-and-Leaf Display If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the display by using two more stems for each leading digit(s). Whenever a stem value is stated twice, the first value corresponds to leaf values of 0-4, and the second values corresponds to values of 5-9. Slide 36
- 37. Example: Hudson Auto Repair Stretched Stem-and-Leaf Display 5 5 6 6 7 7 8 8 9 9 10 10 2 7 2 5 1 5 0 5 1 7 1 5 2 6 1 5 0 8 3 7 4 5 2 7 2 5 2 9 2 8 2 6 3 8 3 7 8 4 8 9 9 9 4 9 9 9 7 8 9 9 Slide 37
- 38. Crosstabulations and Scatter Diagrams Thus far we have focused on methods that are used to summarize the data for one variable at a time. Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables. Crosstabulation and a scatter diagram are two methods for summarizing the data for two (or more) variables simultaneously. Slide 38
- 39. Crosstabulation Crosstabulation is a tabular method for summarizing the data for two variables simultaneously. Crosstabulation can be used when: • One variable is qualitative and the other is quantitative • Both variables are qualitative • Both variables are quantitative The left and top margin labels define the classes for the two variables. Slide 39
- 40. Example: Finger Lakes Homes Crosstabulation The number of Finger Lakes homes sold for each style and price for the past two years is shown below. Price Range < $99,000 > $99,000 Total Home Style Colonial Ranch Split A-Frame Total 18 12 6 14 19 16 12 3 55 45 30 20 35 15 100 Slide 40
- 41. Example: Finger Lakes Homes Insights Gained from the Preceding Crosstabulation • The greatest number of homes in the sample (19) are a split-level style and priced at less than or equal to $99,000. • Only three homes in the sample are an A-Frame style and priced at more than $99,000. Slide 41
- 42. Crosstabulation: Row or Column Percentages Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables. Slide 42
- 43. Example: Finger Lakes Homes Row Percentages Price Range < $99,000 > $99,000 Home Style Colonial Ranch Split A-Frame Total 32.73 26.67 10.91 34.55 31.11 35.56 21.82 6.67 100 100 Note: row totals are actually 100.01 due to rounding. Slide 43
- 44. Example: Finger Lakes Homes Column Percentages Price Range < $99,000 > $99,000 Total Home Style Colonial Ranch Split A-Frame 60.00 40.00 100 30.00 54.29 70.00 45.71 80.00 20.00 100 100 100 Slide 44
- 45. Scatter Diagram A scatter diagram is a graphical presentation of the relationship between two quantitative variables. One variable is shown on the horizontal axis and the other variable is shown on the vertical axis. The general pattern of the plotted points suggests the overall relationship between the variables. Slide 45
- 46. Example: Panthers Football Team Scatter Diagram The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored. x = Number of Interceptions 1 3 2 1 3 y = Number of Points Scored 14 24 18 17 27 Slide 46
- 47. Example: Panthers Football Team Number of Points Scored Scatter Diagram y 30 25 20 15 10 5 0 0 1 2 3 Number of Interceptions x Slide 47
- 48. Example: Panthers Football Team The preceding scatter diagram indicates a positive relationship between the number of interceptions and the number of points scored. Higher points scored are associated with a higher number of interceptions. The relationship is not perfect; all plotted points in the scatter diagram are not on a straight line. Slide 48
- 49. Scatter Diagram A Positive Relationship y x Slide 49
- 50. Scatter Diagram A Negative Relationship y x Slide 50
- 51. Scatter Diagram No Apparent Relationship y x Slide 51
- 52. Tabular and Graphical Procedures Data Qualitative Data Quantitative Data Tabular Methods Graphical Methods Tabular Methods •Frequency Distribution •Rel. Freq. Dist. •% Freq. Dist. •Crosstabulation •Bar Graph •Pie Chart •Frequency Distribution •Rel. Freq. Dist. •Cum. Freq. Dist. •Cum. Rel. Freq. Distribution •Stem-and-Leaf Display •Crosstabulation Graphical Methods •Dot Plot •Histogram •Ogive •Scatter Diagram Slide 52
- 53. End of Chapter 2 Slide 53