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Measures of Dispersion CJ 526 Statistical Analysis in Criminal Justice 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 Six Measures of Variability 1. 2. 3. 4. 5. 6. Variation Ratio Range Interquartile Range Variance Standard Deviation Coefficient of Variation 2. Range Distance between the largest and smallest scores Not very stable—depends on only two scores 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 4. Variance Based on mean squared deviations Deviation = mean minus the score Sum the deviations Squared to eliminate the negative sign Population and Sample Variance 1. Population variance represented by sigma squared: 2 2. Sample variance represented by: s2 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 6. Coefficient of Variation 1. Used to compare SD’s of variables that have different units of measurement (different units of measurement: Height—inches, ACT, 1-36, etc. 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 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 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 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). SPSS Frequencies Procedure Frequencies – Statistics • Central tendency – Mean – Median – Mode SPSS Frequencies Procedure -continued • Dispersion – – – – – – Standard Deviation Variance Range Minimum Maximum Standard Error of the Mean SPSS Descriptives Procedure Descriptives – – – – Minimum Maximum Mean Standard Deviation