1. 1 1 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Chapter-2: Descriptive Statistics: Tabular and Graphical Presentations Part A Exploratory Data Analysis Cross-tabulations; Scatter Diagrams

2. 2 2 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Tabular and Graphical Procedures Qualitative Data Quantitative Data Tabular TabularMethods Methods Methods MethodsGraphical Methods MethodsGraphical Graphical Graphical FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. Percent Freq.Percent Freq. Distribution Distribution CrosstabulationCrosstabulation Bar GraphBar Graph Pie ChartPie Chart FrequencyFrequency Distribution Distribution Rel. Freq. Dist.Rel. Freq. Dist. Cum. Freq. Dist.Cum. Freq. Dist. Cum. Rel. Freq.Cum. Rel. Freq. Distribution Distribution Stem-and-LeafStem-and-Leaf Display Display CrosstabulationCrosstabulation Dot PlotDot Plot HistogramHistogram OgiveOgive ScatterScatter Diagram Diagram DataData

3. 3 3 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Exploratory Data Analysis Exploratory data analysis consist 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. Stem-and-leaf display is one such a technique. Stem-and-leaf display is one such a technique.

4. 4 4 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Stem-and-Leaf Display It is similar to a histogram, but it has the advantage It is similar to a histogram, but it has the advantage of showing the actual data values. A stem-and-leaf display is an exploratory data A stem-and-leaf display is an exploratory data analysis method that shows both the rank order analysis method that shows both the rank order and shape of the distribution of the data. and shape of the distribution of the data.

5. 5 5 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Example: Hudson Auto Repair n Sample of Parts Cost for 50 Tune-ups

6. 6 6 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Stem-and-Leaf Display 5678910 2 7 2 7 2 2 2 2 5 6 7 8 8 8 9 9 9 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 1 1 2 2 3 4 4 5 5 5 6 7 8 9 9 9 0 0 2 3 5 8 9 0 0 2 3 5 8 9 1 3 7 7 7 8 9 1 3 7 7 7 8 9 1 4 5 5 9 1 4 5 5 9

7. 7 7 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Steps to Construct a 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, record the last To the right of the vertical line, record the last digit for each item in rank order. digit for each item in rank order. Arrange the first digits of each data to the Arrange the first digits of each data to the left of a vertical line. left of a vertical line.

8. 8 8 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Stem-and-Leaf Display 5678910 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

9. 9 9 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Stretched Stem-and-Leaf Display Whenever stretched, the first value of a stem Whenever stretched, the first value of a stem corresponds to leaf values of 0  4, and the second corresponds to leaf values of 0  4, and the second value of a stem corresponds to leaf values of 5  9. value of a stem corresponds to leaf values of 5  9. If we believe the display has condensed the data too If we believe the display has condensed the data too much, we can stretch the display by using two stems much, we can stretch the display by using two stems for each leading digit(s ). for each leading digit(s ).

10. 10 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Stretched Stem-and-Leaf Display 5 5 9 1 4 7 7 7 8 9 1 3 5 8 9 0 0 2 3 5 5 5 6 7 8 9 9 9 1 1 2 2 3 4 4 5 6 7 8 8 8 9 9 9 2 2 2 2 7 2 55667788991010

11. 11 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Stem-and-Leaf Display—Some notes 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.

12. 12 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Example: Leaf Unit = 0.1 If we have data with values such as 8 9 10 11 Leaf Unit = 0.1 6 8 1 4 2 0 7 8.6 11.79.49.110.211.08.8 a stem-and-leaf display of these data will be

13. 13 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Example: Leaf Unit = 10 If we have data with values such as 16 17 18 19 Leaf Unit = 10 8 1 9 0 3 1 7 1806171719741791168219101838 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.

14. 14 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Crosstabulations and Scatter Diagrams

15. 15 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Crosstabulations and Scatter Diagrams Crosstabulation and a scatter diagram are two Crosstabulation and a scatter diagram are two such methods for summarizing the data such methods for summarizing the data for two (or more) variables simultaneously. for two (or more) variables simultaneously. Help us understand the relationship between two variables. Help us understand the relationship between two variables. Thus far we have focused on methods that are used Thus far we have focused on methods that are used to summarize the data for one variable at a time. to summarize the data for one variable at a time.

16. 16 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) 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.

17. 17 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) PriceRange Colonial Log Split A-Frame Colonial Log Split A-Frame Total < $99,000 > $99,000 18 6 19 12 5545 30 20 35 15 Total 100 12 14 16 3 Home Style Home Style Cross-tabulation n Example: The number of Homes sold by style and price for the past two years is shown below. quantitative variable variable qualitative

18. 18 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Cross-tabulation n Insights Gained from Preceding Cross-tabulation 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 in the sample (19) The greatest number of homes in the sample (19) 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.

19. 19 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Cross-tabulation: 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.

20. 20 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) PriceRange Colonial Log Split A-Frame Colonial Log Split A-Frame Total < $99,000 > $99,000 18 6 19 12 5545 30 20 35 15 Total 100 12 14 16 3 Home Style Home Style Cross-tabulation Frequency distribution for the price variable Frequency distribution for the home style variable

21. 21 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) PriceRange Colonial Log Split A-Frame Colonial Log Split A-Frame Total < $99,000 > $99,000 32.73 10.91 34.55 21.82 100100 26.67 31.11 35.56 6.67 Home Style Home Style (Colonial and > $99K)/(All >$99K) x 100 = (12/45) x 100 Cross-tabulation: Row Percentages

22. 22 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Simpson’s Paradox Simpson’ Paradox: In some cases the conclusions Simpson’ Paradox: In some cases the conclusions based upon an aggregated cross-tabulation can be based upon an aggregated cross-tabulation can be completely different from the un-aggregated completely different from the un-aggregated 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 cross-tabulation. aggregated cross-tabulation. Often Times data in two or more cross-tabulations are Often Times data in two or more cross-tabulations are aggregated to produce a summary cross-tabulation. aggregated to produce a summary cross-tabulation.

23. 23 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Scatter Diagram and Trend line

24. 24 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) 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 Trend line A trend line is an approximation of the relationship. A trend line is an approximation of the relationship.

25. 25 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Scatter Diagram n A Positive Relationship x y

26. 26 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Scatter Diagram n A Negative Relationship x y

27. 27 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Scatter Diagram n No Apparent Relationship x y

28. 28 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Example: Panthers Football Team n The Panthers football team is interested in investigating the relationship, if any, between interceptions made and points scored. 13213 1424181730 x = Number of Interceptions y = Number of Points Scored Points Scored

29. 29 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) Scatter Diagram y x Number of Interceptions Number of Points Scored 5 10 15 20 25 30 035 12304

30. 30 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) 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 indicates a positive relationship The scatter diagram indicates a positive relationship between the number of interceptions and the between the number of interceptions and the number of points scored. number of points scored. Example: Panthers Football Team

31. 31 Slide © University of Minnesota-Duluth, Summer-2009 Econ-2030(Dr. Tadesse) End of Chapter 2, Part B