1. Dispersion How values arrange themselves around the mean
Applies only to distributions of continuous variables
Includes categorical/ordinal variables that are being treated as continuous
If most scores cluster about the mean the shape of the distribution is peaked
As scores become more dispersed the distribution’s shape flattens

2. Measures of dispersion
Average deviation
Average distance between the mean and the values (scores) for each case
Uses absolute distances (no + or -)
Affected by extreme scores
 (x -x) n
Variance (s2): Squares the numerator – removes negative and is less affected by extreme scores
 (x -x)
n  use n-1 for small samples
Standard deviation (s): Provides results in a manner that is meaningful for the sample being examined
Square root of the variance
Expresses dispersion in units of equal size for that particular distribution
Less affected by extreme scores

3. Demonstration Open DEMO PLUS.sav
Obtain descriptive statistics for continuous variable “age”
Measures of central tendency
Mean, median, mode
Measures of dispersion
Variance and standard deviation
Create a graph (histogram used for continuous variables)
Place a check next to “with normal curve”

4. Practical exercise Enter the scores from this sample into SPSS
Get measures of central tendency, measures of dispersion
Create a graph (histogram)

5. Sample 1 (n=10)
Officer Score Mean Diff. Sq.
Sum
Variance (sum of squares / n-1) s2 .99
Standard deviation (sq. root of variance) s .99

6. Can you do it without the computer?
Sample 2 (n=10)
Officer Score Mean Diff. Sq ___ ___ ___
2 1 ___ ___ ___
3 1 ___ ___ ___
4 2 ___ ___ ___
5 3 ___ ___ ___
6 3 ___ ___ ___
7 3 ___ ___ ___
8 3 ___ ___ ___
9 4 ___ ___ ___
10 2 ___ ___ ___
Sum ____
Variance s2 ____
Standard deviation s ____

7. Sample 2 (n=10)
Officer Score Mean Diff. Sq.
Sum
Variance (sum of squares / n-1) s
Standard deviation (sq. root of variance) s .97

8. Types of Distributions
Unimodal: One hump
Symmetrical: Shapes on both sides of the mean are similar
Mean and standard deviation of a symmetrical, unimodal distribution accurately summarize the underlying data
If distribution is unimodal and symmetrical, can report either median or mean or mode; if not, report all measures

9. Normal curve/normal distribution
A symmetrical, unimodal distribution
About 68 percent of the cases fall within one standard deviation from the mean
About 95 percent fall within two standard deviations from the mean

10. Proportion (percent) of cases under the curve
Z-scores
If the distribution is normal, it is possible to accurately place any score at the point where it would fall within the distribution
To do this must convert the score (x) to a “z” score
Proportion (percent) of cases under the curve
100% of cases

11. Here is a small part of a z-table:
Once a value has been converted to a z-score, a z-score table is used to indicate that value’s position relative to the mean
Since values that are further from the mean occur less frequently, knowing a value’s position provides the probability that it will exist
For example, there is about 1 chance in 10 (.0968) that a value which converts to z = +/- 1.3 will be drawn at random from the distribution
Here is a small part of a z-table: