Mean, Arithmetic

The arithmetic mean is a measure of central tendency, and like all measures of central tendency, it is used to identify a single numerical value that is most “typical” or representative of a data set. Commonly referred to as the “average” or simply the “mean,” the arithmetic mean identifies the numerical average of all values in a data set. It is one of the three most well-known measures of central tendency, along with the median (the middle value) and the mode (the most frequent value), and is the most often used. The arithmetic [Page 933]mean is useful both as a descriptive statistic and as an inferential statistic. In addition, it is used in calculations for many other statistical processes. The arithmetic mean is calculated by adding all values in a data set, then dividing by the number of values in that data set. It is an unbiased calculation of the average, meaning that large outliers and small outliers have the same relative effect on the value of the arithmetic mean. For example, the data set {3, 6, 6, 6, 7, 9, 9, 12, 14} has an arithmetic mean of 8 because

$$3+6+6+6+7+9+9+12+14=72\mathrm{and}\frac{72}{9}=8.$$

By comparison, that data set has a median of 7 because that is the middle value, and it has a mode of 6 because that is the most frequent value. Formally, the arithmetic mean, A, of a set of n numerical data {x1, x2, . . . , xn} is defined as

$$\mathrm{A}=1n_i=1n{x}_i.$$

There are two specific cases of the arithmetic mean that are particularly important in academic research. In the case where a data set includes values for every member of a population, the arithmetic mean is known as the population mean, which is designated by the symbol μ (the lowercase Greek letter mu, pronounced “mew”). In the case where a data set only includes values for selected members of a population, the arithmetic mean is known as a sample mean, which is designated by drawing a bar over the name of the sample variable in question. For example, the sample mean of a data set that includes values for the variable x is designated by $\overline{x}$ (pronounced “x bar”).

This entry examines the arithmetic mean as a descriptive and inferential statistic. This entry also examines how the arithmetic mean can be used in other types of statistical calculations and how it compares to other types of means.