1. Examining Univariate Distributions Chapter 2 SHARON LAWNER WEINBERG SARAH KNAPP ABRAMOWITZ StatisticsSPSS An Integrative Approach SECOND EDITION Using

2. Overview Counting the occurrence of data values Nominal and ordinal (with a limited # of categories) Frequency and percent distribution tables Bar and pie graphs Ordinal (with more than a limited # of categories) and scale variables Frequency and percent distribution tables Histograms and Line graphs Stem-and-leaf displays Accumulating Data Cumulative percent distributions Percentiles and percentile ranks Five number summaries and boxplots

3. Counting the occurrence of data values Categorical variables: Nominal and Ordinal (with fewer than a limited # of values or categories) Frequency and percent distribution tables To create frequency and percent distribution tables, click Analyze on the main menu bar, Descriptive Statistics, and then Frequencies. From the list of variables that appears on the left side of the screen, click the variable, REGION, for example, then click the arrow facing right to move REGION into the box of variables that will be analyzed. Click OK.

4. Example: Frequency Distribution Table for Region from NELS data set

5. Counting the occurrence of data values Categorical variables: Nominal and ordinal (with fewer than a limited number of categories) Bar Graph To generate a bar graph, click Graphs on the main menu bar, Legacy Dialogs, Bar, Define. Move REGION into the Category Axis box and click OK.

6. Example: Bar Graph of Region

7. To edit the graph to include counts, double click to put it in the Chart Editor, click Elements, Show Data Labels.

8. Counting the occurrence of data values Categorical variables: Nominal and ordinal with fewer than a limited # of categories Pie Graph To create a pie graph, click Graphs from the main menu bar, Legacy Dialogs, Pie, Define. Move REGION into the Define Slices by box. Click OK.

9. Example: Pie Graph of Region

10. Counting the occurrence of data values Ordinal (with more than a limited # of categories) and Scale variables Frequency and percentage distribution table SES as an example Reminder: To create a frequency distribution and percent distribution table, click Analyze on the main menu bar, Descriptive Statistics and then Frequencies. Move SES into the Variables Box. Click OK.

11. Example: Frequency Distribution Table for SES

12. Counting the occurrence of data values Ordinal and Scale variables We use graphical summaries that help us understand three features of the distribution: Level Spread Shape

13. Counting the occurrence of data values Ordinal and Scale variables Histogram To obtain the histogram, click Graphs from the main menu bar, Legacy Dialogs, Histogram. Move SES to the Variable box, click OK.

14. Example: Histogram of SES

15. Example: Histogram of Expected Income at Age 30

16. Example: Histogram of Self-Concept in Twelfth Grade

17. Describing a Distribution’s Shape Each of the previous three distributions (SES, Expected Income at Age 30, and Self-Concept in Grade 12) has a different shape. There are terms we may use to describe these different shapes. Symmetric – If, when folded from left to right along its vertical axis of symmetry, one-half of the curve overlaps the other. Asymmetric – If such overlapping does not occur. Skewness – Positive or Negative Which term best describes the shape of each of the previous distributions?

18. Counting the occurrence of data values Ordinal and Scale variables Line graph To obtain the line graph, click Graphs from the main menu bar, Interactive, Line. Move SES to the Horizontal Axis box, and click OK.

19. Example: Line Graph of SES

20. Counting the occurrence of data values Ordinal and Scale variables Stem-and-Leaf To obtain the stem-and-leaf display, click Analyze from the main menu bar, then Descriptive Statistics and then Explore. Move PERTAK to the Dependent List. (If you want to make it so that no extra output is provided, under Display, click the circle next to Plots, click the gray Plots box, and under Boxplots, click the circle next to None.) Click OK.

21. Example: Stem-and-Leaf of Percentage of Students Taking the SAT for the States Data Set

22. Example: Stem-and-Leaf of Average Math SAT for the States Data Set

23. Self-Concept in Twelfth Grade Stem-and-Leaf Plot Frequency Stem & Leaf 5.00 Extremes (=<11) 1.00 1. 3 2.00 1. 55 9.00 1. 666677777 4.00 1. 9999 9.00 2. 000111111 24.00 2. 222222333333333333333333 49.00 2. 4445555555555555555555555555555555555555555555555 52.00 2. 6666677777777777777777777777777777777777777777777777 83.00 2. 88888888888999999999999999999999999999999999999999999999999999999999999999999999999 33.00 3. 000000000111111111111111111111111 42.00 3. 222222222222222333333333333333333333333333 29.00 3. 44444444445555555555555555555 39.00 3. 666666666666666677777777777777777777777 31.00 3. 8888888888888899999999999999999 44.00 4. 00000000000000111111111111111111111111111111 44.00 4. 33333333333333333333333333333333333333333333 Stem width: 10 Each leaf: 1 case(s) Example: Stem-and-Leaf of Twelfth Grade Self-Concept for the NELS Data Set

24. Accumulating Data Accumulating Data (Data are at least ordinal) Tabular: Cumulative percent distribution Graphical: Ogive Percentile rank = % of scores falling below Cumulative percent = % of scores at or below Example: Use the cumulative percent to estimate the percentile rank of an SES value of 13 for the sample of 500 students under study in the NELS data set. Solution: Looking at the last column of the frequency distribution table for SES, the cumulative percent associated with 13 is 25.

25. Accumulating Data Accumulating Data (Data are at least ordinal) Percentiles = score with that % below it. Example: Use SPSS to calculate the 40th percentile and estimate it from the frequency distribution table. The value of the 40 th percentile may be found using SPSS by clicking Analyze, Descriptive Statistics, Frequencies. Move the variable SES to the Variables Box. Click Statistics at the bottom of the dialog box. Click the box next to percentiles and type in 40, Add, Continue. Click OK.

26. Accumulating Data Accumulating Data (Data are at least ordinal) Example: Use SPSS to find the 25th, 50th and 75 percentiles for SES. Then estimate them using the frequency distribution table. The values of the quartiles may be found using SPSS by clicking Analyze, Descriptive Statistics, Frequencies. Move the variable SES to the Variables Box. Click Statistics at the bottom of the dialog box. Click Quartiles, Continue. Click OK.

27. Accumulating Data Accumulating Data (Data are at least ordinal) Boxplot for a single variable To obtain a boxplot for SES, click Graphs, Legacy Dialogs, Boxplot. Click the circle next to Summaries of separate variables, and Define. Move SES to the box labeled Boxes Represent and click OK.

28. Example: Boxplot of SES

29. Example: Boxplot of Twelfth Grade Self-Concept

30. Example: Boxplot of Expected Income at Age 30

31. To obtain a boxplot for SLFCNC08, SLFCNC10, and SLFCNC12, click Graphs, Legacy Dialogs, Boxplot, Summaries of separate variables, and Define. Move SLFCNC08, SLFCNC10, and SLFCNC12 to the box labeled Boxes Represent and click OK. Example: Boxplot of Self-Concept in Eighth, Tenth, and Twelfth Grade for the NELS

32. To create a boxplot of UNITMATH by REGION, click Graphs, Legacy Dialogs, Boxplot, Summaries for Groups of Cases, and Define. Move UNITMATH into the box labeled Variable and REGION into the box labeled Category Axis. Click OK. Example: Boxplot of UNITMATH by REGION for the NELS

33. To obtain a boxplot for SLFCNC08, SLFCNC10, and SLFCNC12 separated by gender, click Graphs, Legacy Dialogs, Boxplot, Clustered, Summaries of separate variables, and Define. Move SLFCNC08, SLFCNC10, and SLFCNC12 to the box labeled Boxes Represent and GENDER to the box labeled category axis and click OK. Example: Boxplot of Self-Concept in Eighth, Tenth, and Twelfth Grade by GENDER for the NELS