Elementary Statistics Chapter 2

1. Elementary Statistics Chapter 2 Frequency Distributions

2. Introduction Descriptive statistics - numbers (or procedures) that describe and summarize an entire set of data. Two basic (and complimentary) approaches. Frequency Distributions - display all of data. “Summary numerical statistics" – identify specific features of a set of data.

3. Frequency Distribution A display of the frequency of occurrence of the values of a variable. One new abbreviation – N stands for the number of observations (scores) in a data set.

4. Simple Frequency Distribution In the simple (regular) frequency distribution, each value is a separate unit in the table. Can be done with both qualitative and quantitative data. To construct a simple frequency distribution: 1. You list all of the categories of your variable.   2. You tally (count) each time that variable has occurred and write frequencies.

5. Simple Frequency Distribution A Set of Qualitative Data See Handout Procedure - List all categories (typically alphabetical but you can do what you want) and tally how many go in each.

6. Simple Frequency Distribution Frequency distributions for quantitative data are constructed in the same way except the presentation order of categories is not arbitrary. Find the highest and the lowest score and write down all of the possible scores in order. Tally (count) each score and convert the tallies to frequencies or relative frequencies. See Handout for Data

7. Relative Frequency Distribution Each frequency count is converted a relative frequency by the following formula: Category Frequency -------------------------- Total Frequency (The total should add to 1).

8. Grouped Frequency Distribution It is often not feasible nor meaningful to include every possible score between the highest and lowest scores in our frequency distribution. We can group our values into equal size class intervals that, taken together, include all of our scores.

9. Grouped Frequency Distribution Example - final grades in a class. 89 78 69 87 A = 90-94 62 76 70 73 B+ = 85-89 73 67 80 73 B = 80-84 77 83 92 75 C+ = 75-79 69 94 86 81 C = 70-74 91 75 78 74 D+ = 65-69 74 71 60 82 D = 60-64 75 67 61 78

10. Comments on Class Intervals  For continuous variables the concept of upper and lower real limits of intervals is relevant. Open-ended frequency distributions (upper or lower class interval is not limited) can used when: 1. There are few scores at the extremes of the distribution. 2. Distinguishing between extreme scores is not meaningful.

11. Comments on Class Intervals Rules when you have to establish your own class intervals (when they are not pre-determined). All intervals must be equal size. The final distribution should be easy to read.

12. Comments on Class Intervals Determine interval size by convenience not just division. Odd numbers make good sizes (because the midpoint a whole number). 5 and 10 make good sizes for graphing. Try to wind up with between 5 and 10 class intervals.

13. Cumulative Frequency Distribution This frequency distribution presents a running tally of scores above or below a given category. Can be done with relative frequencies as well as actual frequencies by converting the numbers to percentages. Standard practice usually has us add from the smallest to the largest.

14. Class Example Sample Data 4 (salary in thousands of dollars) 24 32 43 22 56 78 41 67 50 39 28 36 41 46 30 27 34 46 44 51 62 53 29 32 37 39 43 40 41 52 47 46 51 57 62 55 48 42 29 33 31 36 39 42 42 46 55 61 35 37 48 42 51 21 64

15. Graphing Frequency Distributions General Information. 1. Graphs have an X (horizontal) and a Y (vertical) axis. 2. Your variable is plotted on the X-axis and the frequency counts are plotted on the Y-axis. 3. When graphing quantitative data, the X and Y-axes meet at (0,0).

16. Bar Graph A bar graph is used to graph nominal and USUALLY ordinal data. Procedure 1. Draw your axes and label them. 2. Raise a bar above each variable category to the frequency indicated in your distribution. 3. The bars cannot touch each other.

17. Bar Graph Data on Majors from Tuesday

18. Histogram Histograms are used for discrete or continuous quantitative data (and is preferred for Grouped Frequency Distributions). 1. Constructed like a bar graph except that the bars touch. 2. Good for allowing viewers to find frequencies associated with individual class intervals quickly.

19. Histogram Data on Grades from Tuesday

20. Frequency Polygon Frequency polygons are also used for quantitative data. 1. Draw and label your axes. 2. Put a dot above each variable category indicating the frequency of the category. 3. Connect the dots with a line.

21. Frequency Polygon Sibling Data from Last Time

22. Frequency Polygon 5. When graphing a grouped frequency distribution – plot the midpoint of each class interval (not the upper or lower limits). 6. Good for allowing viewers to see the overall trend of the data. 7. Good for presenting two frequency distributions on a single set of axes.

23. Frequency Polygon Data on Grades (again) from Tuesday

24. Cumulative frequency graphs. To display a running frequency count 1. Draw and label your axes. 2. Put a dot above each variable category indicating the frequency of that category AND all previous categories. 3. Connect the dots with a line.

25. Cumulative frequency graphs. 4. When graphing a grouped frequency distribution – plot the upper real limit of each class interval (not the midpoints).

26. Graphs or Tables? Publications – I prefer tables (people have time to study the data). Presentations – I prefer graphs (people need an overall sense of the data and only have seconds to review it). If it is YOUR data – you get to choose what you want to emphasize.

27. Shapes of Frequency Distributions. Normal Distributions – The shape theoretically associated with all naturally occurring characteristics. Skewed distributions – when there are extreme scores at either the upper or lower end of the distribution.