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

2. Cumulative Frequencies / Percentile Rank # of Arrestsf%cfC% 9-1113.330100 6-8516.62996.5 3-5826.62479.9 0-21653.31653.3 What is the percentile rank for those with 5 arrests?

3. 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%

4. 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

5. Chapter 3 Measures of Central Tendency

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

7. 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

8. 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

9. 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

10. 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

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

12. 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

13. 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

14. 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

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

16. The Mean What does the mean do? – Center of gravity – Deviation = (Raw Score – Mean) X (Raw Score)Deviation 9+3 8+2 60 5 2-4 Mean = 6 = (∑X / N) = (30 / 5)

17. The Weighted Mean The “mean of the means” – overall mean for a number of groups Best used for unequal groups Example: 4, 7, 3, 8 2, 4, 9, 1, 6, 8

18. Obtaining the Mode, Median, and Mean from a Simple Frequency Distribution XfcffX 3112531 3012430 2912329 280220 2722254 2632078 2511725 2411624 2321546 2221344 2121142 203960 194676 182236 Position of the Mdn Mo Mdn

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. 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

21. Level of Measurement

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