1. Measures of Dispersion
CJ 526 Statistical Analysis in Criminal Justice

2. Introduction
Variability provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together
All measurements vary. If there were no variation, one unit could be measured, and there would be no need for other measurement or statistics

3. Six Measures of Variability
Variation Ratio
Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation

4. 2. Range Distance between the largest and smallest scores
Not very stable—depends on only two scores

5. 3. Interquartile Range
The distance between the first quartile and third quartile
The score at the 75th percentile minus the score at the 25th percentile
50% of the scores fall in this range, 25% above this range and 25% below this range

6. 4. Variance Based on mean squared deviations
Deviation = mean minus the score
Sum the deviations
Squared to eliminate the negative sign

7. Population and Sample Variance
Population variance represented by sigma squared: 2
Sample variance represented by: s2

8. 5. Standard Deviation
Standard deviation is the square root of variance
More easily interpreted than the variance, more meaningful
Standard deviation for a sample represented by the symbol SD

9. 6. Coefficient of Variation
Used to compare SD’s of variables that have different units of measurement
(different units of measurement: Height—inches, ACT, 1-36, etc.

10. Interpretation of SD SD used with the mean, must be interval level
Add and subtract SD from the Mean
i.e., if the Mean is 10 and the SD is 2, add it to the mean (12) and subtract it from the mean (8)
If the mean is 10 and the SD is 2, the majority of the scores fall in the range between 8 and 12
If the SD is 4, most of the scores are between 6 and 14

11. Interpretation of SD
Generally speaking, the larger the SD, the more variability in that sample for that variable
The smaller the SD, the less variability
If one sample had a SD of 2 and another of 4, for a particular variable, the one with an SD of 2 had less variability

12. Interpretation of SD
A particular variable can be compared from sample to sample in terms of variability
However, if the variables are on different scales, samples cannot be compared in terms of variability using SD (z scores would be needed, which will be covered later)
Example: the SAT has a SD of 100, IQ has an SD of 15. Variability cannot be compared, as they are different scales

13. Report Writing - Text
Children who viewed the violent cartoon displayed more aggressive responses (M = 12.45, SD = 3.7) than those who viewed the control cartoon (M = 4.22, SD = 1.04).

14. SPSS Frequencies Procedure
Statistics
Central tendency
Mean
Median
Mode

15. SPSS Frequencies Procedure -- continued
Dispersion
Standard Deviation
Variance
Range
Minimum
Maximum
Standard Error of the Mean

16. SPSS Descriptives Procedure
Minimum
Maximum
Mean
Standard Deviation