1. Chapter 2 Review Using graphs/tables/diagrams to show variable relationships Understand cumulative frequency/percentage and cross-tabulations Perform rates of change

2. Cumulative Frequencies # of Arrests f cf %C% 9-111 6-85 3-58 0-216

3. Cumulative Frequencies # of Arrests f cf %C% 9-111303.3100 6-852916.696.5 3-582426.679.9 0-216 53.3 N = 30

4. Cross-Tabulations Attitude towards Lowering the Drinking Age to 19 MaleFemaleTotal Favor5847651349 Neutral4989021400 Oppose108216672749 Total216433345498

5. Cross-Tabulations Attitude towards Lowering the Drinking Age to 19 MaleFemaleTotal Favor5847651349 27%23%24.5% Neutral4989021400 23%27%25.5% Oppose108216672749 50% Total216433345498 100%

6. Rate of Change Rate of Change = (100) * (time 2f – time 1f) (time1f) Allows us to compare the same population at two points in time. Always be aware of the sign. – A negative percent signifies a reduction – A positive percent signifies an increase

7. In 2011, there were 60 thefts from a building. In 2010, there were 40. What is the rate of change? In 2010, there were 51 cases of forcible rape whereas in 2011, there was 35. What is the rate of change?

8. Chapter 3 Measures of Central Tendency

9. Measures of Central Tendency Three main types – Mode – Median – Mean Choice depends upon level of measurement

10. The Mode The mode is the most frequently occurring value in a distribution. Abbreviated as Mo Sometimes there is more than one mode EX: 96, 91, 96, 90, 93, 90, 96, 90 Bimodal Mode is the only measure of central tendency appropriate for nominal-level variables

11. Mode - Example What is the mode for the following set of numbers? 20, 21, 30, 20, 22, 20 Explains nothing about – Ordering of variables – Variation within variables Distributions can be bimodal and/or multimodal – Several categories with same frequencies

12. The Median The median is the middle case of a distribution Abbreviated as Mdn Appropriate for ordinal data because it only shows direction and not distance Used if distribution is skewed How to find the median? If even, there will be two middle cases – interpolate If odd, choose the middle-most case Cases must be ordered Position of the Mdn

13. Example of median: Years in Prison What is the median? – odd or even? (7+1)/2=4th case Where is the 4th case? Sort distribution from lowest to highest 1 5 2 9 13 11 4

14. Example of median: Years in Prison 4th case 1 2 4 5* 9 11 13

15. Example of median with even # of cases (8+1)/2=4.5 Half way between the 4th and 5th case (2 + 3) / 2 = 2.5 Median = 2.5 1 2 3 4 6 Position of the Mdn

16. The Mean Most popular measure of central tendency Assumes equality of intervals Basis of many higher order formulas for statistical procedures Use either μ or X depending on whether population or sample estimate

17. The Mean The mean is appropriate for interval and ratio level variables X = raw scores in a set of scores N = total number of scores in a set

18. Example: Prison Sentences What is the mean? 4.6 7.9 11.4 2.2

19. An Illustration: Measures of Central Tendency in a Skewed Distribution Salary $120,000 $60,000 $40,000 $30,000  Mean = $50,000  Median = $40,000  Mode = $30,000

20. Level of Measurement

21. Comparing the Mode, Median, and Mean Three factors in choosing a measure of central tendency 1.Level of measurement 2.Shape or form of the distribution of data Skewness Kurtosis 3.Research Objective

22. Shape of the Distribution In symmetrical distribution – mode, median, and mean have identical values In skewed data, the measures of central tendency are different – Skewness relevant only at the interval level Mean heavily influenced by extreme outliers – median best measure in this situation

24. Research Objective Choice of reported central tendency depends on the level of precision required. Most published research requires median and/or mean calculations. In skewed data, median more balanced view For advanced statistical analyses, mean usually preferred In large data sets, mean most stable measure

25. Summary Three best known measure of central tendency – mode, median, mode Three factors determine appropriateness – Level of measurement – Shape of the distribution – Research objective