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

Slides Prepared by JOHN S. LOUCKS St. Edward’s University. y. x. Chapter 2 Descriptive Statistics: Tabular and Graphical Presentations Part B. Exploratory Data Analysis Crosstabulations and Scatter Diagrams. Minor mods by McKim. Exploratory Data Analysis.

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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. y x Chapter 2Descriptive Statistics:Tabular and Graphical PresentationsPart B • Exploratory Data Analysis • Crosstabulations and Scatter Diagrams Minor mods by McKim

  3. 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.

  4. 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.

  5. Example: Hudson Auto Repair 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.

  6. Example: Hudson Auto Repair • Sample of Parts Cost for 50 Tune-ups

  7. Stem-and-Leaf Display 5 6 7 8 9 10 2 7 2 2 2 2 5 6 7 8 8 8 9 9 9 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 0 0 2 3 5 8 9 1 3 7 7 7 8 9 1 4 5 5 9 a stem a leaf

  8. 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 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 • value corresponds to leaf values of 5 - 9.

  9. Stretched Stem-and-Leaf Display 5 5 6 6 7 7 8 8 9 9 10 10 2 7 2 2 2 2 5 6 7 8 8 8 9 9 9 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 0 0 2 3 5 8 9 1 3 7 7 7 8 9 1 4 5 5 9

  10. 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.

  11. 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 9 10 11 6 8 1 4 2 0 7

  12. 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 17 18 19 8 The 82 in 1682 is rounded down to 80 and is represented as an 8. 1 9 0 3 1 7

  13. 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.

  14. Crosstabulation • A crosstabulation is a tabular summary of data for two variables. • Crosstabulation can be used when: • one variable is qualitative and the other is • quantitative, • both variables are qualitative, or • both variables are quantitative. • In other words, any time!! (Duh.) • The left and top margin labels define the classes for • the two variables.

  15. Crosstabulation • Example: Finger Lakes Homes The number of Finger Lakes homes sold for each style and price for the past two years is shown below. qualitative variable quantitative variable Home Style Price Range Colonial Log Split A-Frame Total 18 6 19 12 55 45 < $99,000 > $99,000 12 14 16 3 30 20 35 15 Total 100

  16. Crosstabulation • Insights Gained from 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.

  17. Crosstabulation Frequency distribution for the price variable Home Style Price Range Colonial Log Split A-Frame Total 18 6 19 12 55 45 < $99,000 > $99,000 12 14 16 3 30 20 35 15 Total 100 Frequency distribution for the home style variable

  18. 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.

  19. Crosstabulation: Row Percentages Home Style Price Range Colonial Log Split A-Frame Total 32.73 10.91 34.55 21.82 100 100 < $99,000 > $99,000 26.67 31.11 35.56 6.67 Note: row totals are actually 100.01 due to rounding. (Colonial and > $99K)/(All >$99K) x 100 = (12/45) x 100

  20. Crosstabulation: Column Percentages Home Style Price Range Colonial Log Split A-Frame 60.00 30.00 54.29 80.00 < $99,000 > $99,000 40.00 70.00 45.71 20.00 100 100 100 100 Total (Colonial and > $99K)/(All Colonial) x 100 = (12/30) x 100

  21. Crosstabulation: Simpson’s Paradox • Data in two or more crosstabulations are often aggregated to produce a summary crosstabulation. • We must be careful in drawing conclusions about the • relationship between the two variables in the • aggregated crosstabulation. • Simpson’ Paradox: In some cases the conclusions based upon an aggregated crosstabulation can be completely reversed if we look at the unaggregated data. suggests the overall relationship between the variables. (Yikes, just ignore that last partial sentence.)

  22. Simpson’s Paradox (p.48) Judge Verdict Total Appealed Luckett Kendall 129 (86%) 110 (88%) Upheld Reversed 239 36 • 21 (14%) 15 (12%) 150 (100%) 125 (100%) Total (%) 275 Kendall appears to have the better rate of upheld verdicts

  23. Simpson’s Paradox (Cont’d) Judge Luckett Verdict Total Appealed Common Pleas Municipal 29 (91%) 100 (85%) Upheld Reversed 129 21 • 3 (9%) 18 (15%) 32 (100%) 118 (100%) Total (%) 150

  24. Simpson’s Paradox (Cont’d) Judge Kendall Verdict Total Appealed Common Pleas Municipal 90 (90%) 20 (80%) Upheld Reversed 110 15 • 10 (10%) 5 (20%) 100 (100%) 25 (100%) Total (%) 125

  25. Simpson’s Paradox (Cont’d) Luckett has the better rate of upheld verdicts in both Common Pleas and Municipal Courts Yet the aggregate table makes it appear the Kendall has the better rate. What appears to have happened is that Municipal Court appeals have a higher overturn rate and Luckett just happened to try most of his cases there. What other important factor is missing from this analysis??

  26. Scatter Diagram and Trendline • 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. • A trendline is an approximation of the relationship. • Example in book is number of commercials vs sales

  27. Scatter Diagram • A Positive Relationship y x

  28. Scatter Diagram • A Negative Relationship y x

  29. Scatter Diagram • No Apparent Relationship y x

  30. 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 y = Number of Points Scored 14 24 18 17 30 1 3 2 1 3

  31. 35 30 25 20 15 10 5 0 1 0 2 3 4 Scatter Diagram y Number of Points Scored x Number of Interceptions

  32. Example: Panthers Football Team • Insights Gained from the Preceding Scatter Diagram • The 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.

  33. Tabular and Graphical Procedures Data Qualitative Data Quantitative Data Tabular Methods Graphical Methods Tabular Methods Graphical Methods • Bar Graph • Pie Chart • Frequency • Distribution • Rel. Freq. Dist. • Percent Freq. • Distribution • Crosstabulation • Dot Plot • Histogram • Ogive • Scatter • Diagram • Frequency • Distribution • Rel. Freq. Dist. • Cum. Freq. Dist. • Cum. Rel. Freq. • Distribution • Stem-and-Leaf • Display • Crosstabulation

  34. Keep in mind… • Frequency Distribution shows how many. • Rel. Freq. Distribution shows what fraction. • Percent Freq. Distribution shows what percentage. • Can represent any of the above in tables or charts.

  35. End of Chapter 2, Part B

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