# Mean, The

# Mean, The

The mean is widely used throughout the social sciences and is the arithmetic average of a set of scores. It is simply the sum of all the scores of interest divided by the number of scores. In casual conversation this calculation is referred to as “getting the average.” The capital letter *M* is commonly used in journal articles to represent the mean. The formula for the mean is quite simple: *M* = *ΣX/N*, where *M* is the mean, *ΣX* is the sum of all the scores of interest, and *N* is the number of the scores.

In general, what is sought is a single value that represents the location of the set of scores on some scale. The mean describes the central tendency of a distribution of scores. In other words, it describes where the scores tend to cluster together or the arithmetic middle point of the scores. This measure of central tendency is sometimes referred to as the index of location.

The mode and the median are also used to describe central tendency; they lack, however, the mathematical properties of the mean. These other measures are most useful when the distribution of scores is multimodal or skewed.

A simple example will illustrate the mean. Assume one has five scores: 6, 5, 4, 3, and 2. The sum of these scores is 20. The mean is 20/5, which equals 4. In this case the median is also 4 (it is the score that divides all the scores in half). This is usually the case when the distribution of scores is symmetrical. This example can also illustrate an advantage of the mean over the median. The mean is sensitive to all the scores in the distribution, while the median is not. Look at what happens if the score of 6 is changed to a score of 16. The new sum of the scores is now 30, and the mean is 30/5, which is 6. But the median is still 4. The mean reflects the value of every score in the distribution. Of course, in the social sciences one often encounters distributions that are essentially normal, in which case all three measures of central tendency (mode, median, and mean) will be the same.

For many variables in the social sciences the mean is the standard descriptor of central tendency. For example, the mean may be used to describe a person’s average grade in a course of study or a person’s average weight over a given period. The mean is useful in describing groups as well. Demographic data are often presented as means (e.g., mean age, height, weight, number of children, or siblings). The average performance of a class of students, a school, a school district, a state, or the whole country may be reported on some measure of academic achievement. Even countries may be compared on average academic skills.

In research or evaluation the characteristics of a population of interest often need to be described. Populations, however, are typically very large (sometimes infinitely so) and impossible to measure completely. The truth about a population may only be inferred. For this reason a sample is collected from the population, with the intent of generalizing the results to the whole population. Once the sample scores have been obtained, the data need to be described in such a way as to communicate the essential characteristics and also allow inferences about the population—briefly but informatively.

The sample mean is an unbiased estimator of the population mean. It determines a center point of the set of scores that includes the value of every score in the set. Because the mean includes all the scores, it is the point of balance or center of gravity of the distribution.

Much research is concerned with trying to determine how different or how similar people or groups may be. The mean is used to do this. The measurement of how far individuals are from their group mean is taken, and also the measurement of how far group means are from the mean of all the samples. The resulting scores are called deviation scores, and are obtained by subtracting the mean from the individual score. These deviation scores can be used to calculate another average called the variance (because the sum of the deviation scores is always zero, they must be squared before being added together and dividing by the number). The square root of the variance is called the standard deviation. This average tells one about the amount of variability in a set of scores or among a set of means.

This property of the mean is critical because much research is concerned with examining how much sample means deviate from some point or from each other. If the means deviate from each other by a large enough amount, they are said to be “significantly different.” This is the basis for significance testing between groups. These significance tests are commonly found in published research, although there is debate about which approach to use and how results should be reported.

The mean is arguably the most widely used measure of central tendency. It requires interval or ratio level of measurement, works best with distributions that are unimodal and roughly symmetrical, and is the basis for much statistical decision-making.

**SEE ALSO** *Decision-making; Population Studies; Standard Deviation*

## BIBLIOGRAPHY

Aron, Arthur, and Elaine N. Aron. 2003. *Statistics for Psychology*. 3rd ed. Upper Saddle River, NJ: Prentice Hall.

Hays, William L. 1973. *Statistics for the Social Sciences*. 2nd ed. New York: Holt, Rinehart and Winston.

Oaks, Michael W. 1986. *Statistical Inference: A Commentary for the Social and Behavioural Sciences*. Chichester, U.K.: Wiley.

*Samuel K. Rock Jr.*

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