Understanding Variable Relationships and Measures of Central Tendency in Data Analysis

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 What is the percentile rank for those with 5 arrests?

3. Cross-Tabulations Attitude towards Lowering the Drinking Age to 19

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 3Measures 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. Position of the Mdn 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

10. 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 Example of median: Years in Prison

11. (8+1)/2=4.5 Half way between the 4th and 5th case (2 + 3) / 2 = 2.5 Median = 2.5 1 1 2 2 3 4 4 6 Example of median with even # of cases Position of the Mdn

12. 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

13. 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

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

15. The Mean • What does the mean do? • Center of gravity • Deviation = (Raw Score – Mean) Mean = 6 = (∑X / N) = (30 / 5)

16. 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

17. An Illustration: Measures of Central Tendency in a Skewed Distribution • Mean = $50,000 • Median = $40,000 • Mode = $30,000

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

19. Level of Measurement

20. 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

21. 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

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