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

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)2 ----------- 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

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”

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

Sample 1 (n=10) Officer Score Mean Diff. Sq. 1 3 2.9 .1 .01 2 3 2.9 .1 .01 3 3 2.9 .1 .01 4 3 2.9 .1 .01 5 3 2.9 .1 .01 6 3 2.9 .1 .01 7 3 2.9 .1 .01 8 1 2.9 1.9 3.61 9 2 2.9 .9 .81 10 5 2.9 2.1 4.41 Sum 8.90 Variance (sum of squares / n-1) s2 .99 Standard deviation (sq. root of variance) s .99

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

Sample 2 (n=10) Officer Score Mean Diff. Sq. 1 2 2.4 .4 .16 2 1 2.4 1.4 1.96 3 1 2.4 1.4 1.96 4 2 2.4 .4 .16 5 3 2.4 .6 .36 6 3 2.4 .6 .36 7 3 2.4 .6 .36 8 3 2.4 .6 .36 9 4 2.4 1.6 2.56 10 2 2.4 .4 .16 Sum 8.40 Variance (sum of squares / n-1) s2 .93 Standard deviation (sq. root of variance) s .97

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

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

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

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: