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Type of average of a collection of numbers

In mathematics and statistics, the arithmetic mean (/pron/,/æ//r//ɪ//θ/'/m//ɛ//t//ɪ//k// /'/m//iː//n/ arrithMETik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic. In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.

The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include:

If numbers have mean , then Since is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which the residuals (deviations from the estimate) sum to zero. This can also be interpreted as saying that the mean is translationally invariant in the sense that for any real number , If it is required to use a single number as a "typical" value for a set of known numbers , then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of The sample mean is also the best single predictor because it has the lowest root mean squared error. If the arithmetic mean of a population of numbers is desired, then the estimate of it that is unbiased is the arithmetic mean of a sample drawn from the population. The arithmetic mean is independent of scale of the units of measurement, in the sense that So, for example, calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean. This is also called first order homogeneity.

The arithmetic mean of a sample is always between the largest and smallest values in that sample. The arithmetic mean of any amount of equal sized number groups together is the arithmetic mean of the arithmetic means of each group.

The arithmetic mean may be contrasted with the median. The median is defined such that no more than half the values are larger, and no more than half are smaller than it. If elements in the data increase arithmetically when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample The mean is , as is the median. However, when we consider a sample that cannot be arranged to increase arithmetically, such as , the median and arithmetic average can differ significantly. In this case, the arithmetic average is , while the median is The average value can vary considerably from most values in the sample and can be larger or smaller than most. There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income.

A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation. For example, the arithmetic mean of and is , or equivalently In contrast, a weighted mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as Here the weights, which necessarily sum to one, are and , the former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number ( in the above example and in a situation with numbers being averaged).

If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero. In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the mean of the probability distribution. The most widely encountered probability distribution is called the normal distribution; it has the property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms), are equal. This equality does not hold for other probability distributions, as illustrated for the log normal distribution here.

The arithmetic mean is often denoted by a bar (vinculum or macron), as in
In some document formats (such as PDF), the symbol may be replaced by a "¢" (cent) symbol when copied to a text processor such as Microsoft Word.