Middle Tennessee State University Social Work Department

Data Analysis for Social Work
Lecture Handout 7

## Measures of Dispersion

Measures of Dispersion are descriptive statistics which provide information about the extent to which the responses from different elements are spread out across the variable's values. These are used to determine the degree of diversity among respondents.

One way to think about dispersion is reflected in two terms which are used frequently among researchers, especially in reference to ordinal and interval data. These terms are:

1. Peaked distributions - which are frequency distributions with less dispersion or variability. There are many people with the same haracteristics. The graphs of these distributions have a "peak."

2. Flat distributions - which have more variability. People are more spread out with respect to different attributes. The graphs are "flat."

Measures of dispersion include:

A. Range - a measure of dispersion which is based on the total distance between the highest and lowest values. This is calculated by subtracting the number which represents the lowest value from the number of the highest value. The more categories which had actual answers from elements, the higher the range. This works for all levels of measurement.

For example, the range for gender is always 0 or 1. A range of zero would mean that all elements are the same gender. A range of one would mean both genders are represented in the sample or population. (2-1=1)

The age range would have a low value for a homogenous sample (20-18=2); a high value for a sample with people who were very different in age (87-14=73).

B. Dispersion (S) - a measure of how far from a even spread the values of a variable are. S varies between 0 and 1, with zero meaning no dispersion; one, a completely even spread. This may be used for all levels of measurement.

One formula for dispersion is:
S = 1 -
 K(M) N(K-1)

where K is the total number of values, M is the number of cases that have to move for an even spread, and N is the number of cases. These formulas are relatively new, not used very frequently, and not included in the SPSS statistics. (You are not required to remember the formula.)

C. Standard Deviation(s) - the amount an average score differs from the mean. This is appropriate only for interval or dichotomous variables. The higher the standard deviation, the more diverse the scores. This is a very popular measure of dispersion, is produced by SPSS, and is sometimes abused through its use with ordinal data.

The formula is: s = the square root of the sum of the difference between each score and the mean squared divided by the number of scores.
(You are not required to remember the formula.)

For example, there are two samples of two people. The first sample is composed of one person who makes \$1,000,000. and one who makes nothing. The mean income for this sample is \$500,000. The standard deviation is: 500,000

The second sample is composed of two people who both make \$500,000. The mean is still the same, \$500,000., but the standard deviation is zero.

D. Z score - mathematical way to relate a specific score to the mean and standard deviation. This is a way to tell how different an individual score is from the rest of the scores. The formula is:

Z = the specific score minus the mean divided by the standard deviation (You are not required to remember this.)