# Normal Distribution

The common pattern of numbers in which the majority of the measurements tend to cluster near the mean of distribution.

An example of the bell-shaped curve of a normal distribution.

Psychological research involves measurement of behavior. This measurement results in numbers that differ from one another individually but that are predictable as a group. One of the common patterns of numbers involves most of the measurements being clustered together near the mean of the distribution, with fewer cases occurring as they deviate farther from the mean. When a frequency distribution is drawn in pictorial form, the resulting pattern produces the bell-shaped curve that scientists call a normal distribution.

When measurements produce a normal distribution, certain things are predictable. First, the mean, median, and mode are all equal. Second, a scientist can predict how far from the mean most scores are likely to fall. Thus, it is possible to determine which scores are more likely to occur and the proportion of score likely to be above or below any given score.

Many behavioral measurements result in normal distributions. For example, scores on intelligence tests are likely to be normally distributed. The mean is about 100 and a typical person is likely to score within about 15 points of the mean, that is, between 85 and 115. If the psychologist knows the mean and the typical deviation from the mean (called the standard deviation), the researcher can determine what proportion of scores is likely to fall in any given range. For instance, in the range between one standard deviation below the mean (about 85 for IQ scores) and one deviation above the mean (about 115 for IQ scores), one expects to find the scores of about two thirds of all test takers. Further, only about two and a half percent of test takers will score higher than two standard deviations above the mean (about 130).

Although psychologists rely on the fact that many measurements are normally distributed, there are certain cases where scores are unlikely to be normally distributed. Whenever scores cannot be higher than some upper value or smaller than some lower value, a non-normal distribution may occur. For example, salaries are not normally distributed because there is a lower value (i.e., nobody can make less than zero dollars), but there is no upper value. Consequently, there will be some high salaries that will not be balanced by corresponding, lower salaries. It is important to know whether scores are normally distributed because it makes a difference in the kind of statistical tests that are appropriate for analyzing and interpreting the numbers.