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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
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