1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 2, Part B Descriptive Statistics: Tabular and Graphical Presentations n Exploratory Data Analysis: Stem-and-Leaf Display n Crosstabulations and Scatter Diagrams Scatter Diagrams x y

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Exploratory Data Analysis The techniques of exploratory data analysis consist of The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly. be used to summarize data quickly. One such technique is the stem-and-leaf display. One such technique is the stem-and-leaf display.

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Stem-and-Leaf Display Each digit on a stem is a leaf. Each digit on a stem is a leaf. Each line in the display is referred to as a stem. Each line in the display is referred to as a stem. To the right of the vertical line we record the last To the right of the vertical line we record the last digit for each item in rank order. digit for each item in rank order. The first digits of each data item are arranged to the The first digits of each data item are arranged to the left of a vertical line. left of a vertical line. It is similar to a histogram on its side, but it has the It is similar to a histogram on its side, but it has the advantage of showing the actual data values. advantage of showing the actual data values. A stem-and-leaf display shows both the rank order A stem-and-leaf display shows both the rank order and shape of the distribution of the data. and shape of the distribution of the data.

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved 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 6 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Hudson Auto Repair n Sample of Parts Cost ($) for 50 Tune-ups

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Stem-and-Leaf Display a stem a leaf

8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Stretched Stem-and-Leaf Display Whenever a stem value is stated twice, the first value Whenever a stem value is stated twice, the first value corresponds to leaf values of 0  4, and the second corresponds to leaf values of 0  4, and the second value corresponds to leaf values of 5  9. value corresponds to leaf values of 5  9. If we believe the original stem-and-leaf display has If we believe the original stem-and-leaf display has condensed the data too much, we can stretch the condensed the data too much, we can stretch the display by using two stems for each leading digit(s). display by using two stems for each leading digit(s).

9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Stretched Stem-and-Leaf Display

10 Slide © 2008 Thomson South-Western. All Rights Reserved Stem-and-Leaf Display n Leaf Units Where the leaf unit is not shown, it is assumed Where the leaf unit is not shown, it is assumed to equal 1. to equal 1. Leaf units may be 100, 10, 1, 0.1, and so on. Leaf units may be 100, 10, 1, 0.1, and so on. In the preceding example, the leaf unit was 1. In the preceding example, the leaf unit was 1. A single digit is used to define each leaf. A single digit is used to define each leaf.

11 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Leaf Unit = 0.1 If we have data with values such as Leaf Unit = a stem-and-leaf display of these data will be

12 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Leaf Unit = 10 If we have data with values such as Leaf Unit = a stem-and-leaf display of these data will be The 82 in 1682 is rounded down to 80 and is represented as an 8. The 82 in 1682 is rounded down to 80 and is represented as an 8.

13 Slide © 2008 Thomson South-Western. All Rights Reserved Crosstabulations and Scatter Diagrams Crosstabulation and a scatter diagram are two Crosstabulation and a scatter diagram are two methods for summarizing the data for two variables methods for summarizing the data for two variables simultaneously. simultaneously. Often a manager is interested in presentations that Often a manager is interested in presentations that will help understand the relationship between two will help understand the relationship between two variables. variables. Thus far we have focused on presentations that are Thus far we have focused on presentations that are used to summarize the data for one variable at a time. used to summarize the data for one variable at a time.

14 Slide © 2008 Thomson South-Western. All Rights Reserved Crosstabulation The left and top margin labels define the classes for The left and top margin labels define the classes for the two variables. the two variables. n Crosstabulation can be used when: one variable is qualitative and the other is one variable is qualitative and the other is quantitative, quantitative, both variables are qualitative, or both variables are qualitative, or both variables are quantitative. both variables are quantitative. A crosstabulation is a tabular summary of data for A crosstabulation is a tabular summary of data for two variables. two variables.

15 Slide © 2008 Thomson South-Western. All Rights Reserved Price Range Colonial Log Split A-Frame Total < $99,000 > $99, Total Home Style Home Style Crosstabulation n Example: Finger Lakes Homes The number of Finger Lakes homes sold for each style and price for the past two years is shown below. quantitative variable variablequantitative qualitative qualitative

16 Slide © 2008 Thomson South-Western. All Rights Reserved Crosstabulation n Insights Gained from Preceding Crosstabulation Only three homes in the sample are an A-Frame Only three homes in the sample are an A-Frame style and priced at more than $99,000. style and priced at more than $99,000. The greatest number of homes (19) in the sample The greatest number of homes (19) in the sample are a split-level style and priced at less than or are a split-level style and priced at less than or equal to $99,000. equal to $99,000.

17 Slide © 2008 Thomson South-Western. All Rights Reserved PriceRange Colonial Log Split A-Frame Colonial Log Split A-Frame Total < $99,000 > $99, Total Home Style Home Style Crosstabulation Frequency distribution for the price variable Frequency distribution for the home style variable

18 Slide © 2008 Thomson South-Western. All Rights Reserved Crosstabulation: Row or Column Percentages n Converting the entries in the table into row percentages or column percentages can provide additional insight about the relationship between the two variables.

19 Slide © 2008 Thomson South-Western. All Rights Reserved PriceRange Colonial Log Split A-Frame Colonial Log Split A-Frame Total < $99,000 > $99, Note: row totals are actually due to rounding Home Style Home Style (Colonial and > $99K)/(All >$99K) x 100 = (12/45) x 100 Crosstabulation: Row Percentages

20 Slide © 2008 Thomson South-Western. All Rights Reserved PriceRange Colonial Log Split A-Frame Colonial Log Split A-Frame < $99,000 > $99, Home Style Home Style Total (Colonial and > $99K)/(All Colonial) x 100 = (12/30) x 100 Crosstabulation: Column Percentages

21 Slide © 2008 Thomson South-Western. All Rights Reserved Crosstabulation: Simpson’s Paradox Simpson’ Paradox: In some cases the conclusions Simpson’ Paradox: In some cases the conclusions based upon an aggregated crosstabulation can be based upon an aggregated crosstabulation can be completely reversed if we look at the unaggregated completely reversed if we look at the unaggregated data. data. We must be careful in drawing conclusions about the We must be careful in drawing conclusions about the relationship between the two variables in the relationship between the two variables in the aggregated crosstabulation. aggregated crosstabulation. Data in two or more crosstabulations are often Data in two or more crosstabulations are often aggregated to produce a summary crosstabulation. aggregated to produce a summary crosstabulation.

22 Slide © 2008 Thomson South-Western. All Rights Reserved The general pattern of the plotted points suggests the The general pattern of the plotted points suggests the overall relationship between the variables. overall relationship between the variables. One variable is shown on the horizontal axis and the One variable is shown on the horizontal axis and the other variable is shown on the vertical axis. other variable is shown on the vertical axis. A scatter diagram is a graphical presentation of the A scatter diagram is a graphical presentation of the relationship between two quantitative variables. relationship between two quantitative variables. Scatter Diagram and Trendline A trendline is an approximation of the relationship. A trendline is an approximation of the relationship.

23 Slide © 2008 Thomson South-Western. All Rights Reserved n A Positive Relationship x y Scatter Diagram and Trendline

24 Slide © 2008 Thomson South-Western. All Rights Reserved n A Negative Relationship x y Scatter Diagram and Trendline

25 Slide © 2008 Thomson South-Western. All Rights Reserved n No Apparent Relationship x y Scatter Diagram and Trendline

26 Slide © 2008 Thomson South-Western. All Rights Reserved Example: Panthers Football Team n Scatter Diagram and Trendline The Panthers football team is interested 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 Points Scored

27 Slide © 2008 Thomson South-Western. All Rights Reserved y x Number of Interceptions Number of Points Scored Scatter Diagram and Trendline

28 Slide © 2008 Thomson South-Western. All Rights Reserved n Insights Gained from the Preceding Scatter Diagram The relationship is not perfect; all plotted points in The relationship is not perfect; all plotted points in the scatter diagram are not on a straight line. the scatter diagram are not on a straight line. Higher points scored are associated with a higher Higher points scored are associated with a higher number of interceptions. number of interceptions. The scatter diagram and trendline indicate a The scatter diagram and trendline indicate a positive relationship between the number of positive relationship between the number of interceptions and the number of points scored. interceptions and the number of points scored. Example: Panthers Football Team

29 Slide © 2008 Thomson South-Western. All Rights Reserved Tabular and Graphical Procedures Qualitative Data Quantitative Data Tabular TabularMethods Methods Methods MethodsGraphical Methods MethodsGraphical Graphical Graphical Frequency Frequency Distribution Distribution Relative Freq. Relative Freq. Distribution Distribution Percent Freq. Percent Freq. Distribution Distribution Crosstabulation Crosstabulation Bar Graph Bar Graph Pie Chart Pie Chart Frequency Dist. Frequency Dist. Rel. Freq. Dist. Rel. Freq. Dist. % Freq. Dist. % Freq. Dist. Cum. Freq. Dist. Cum. Freq. Dist. Cum. Rel. Freq. Cum. Rel. Freq. Distribution Distribution Cum. % Freq. Cum. % Freq. Distribution Distribution Crosstabulation Crosstabulation Dot Plot Dot Plot Histogram Histogram Ogive Ogive Stem-and- Stem-and- Leaf Display Leaf Display Scatter Scatter Diagram Diagram DataData

30 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 2, Part B