Arithmetic mean

In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. (The word set is used perhaps somewhat loosely; for example, the number 3.8 could occur more than once in such a "set".) The arithmetic mean is what pupils are taught very early to call the "average." If the set is a statistical population, then we speak of the population mean. If the set is a statistical sample, we call the resulting statistic a sample mean.

The mean may be conceived of as an estimate of the median. When the mean is not an accurate estimate of the median, the set of numbers, or frequency distribution, is said to be skewed.

We denote the set of data by X = {x1, x2, ..., xn}. The symbol μ (Greek: mu) is used to denote the arithmetic mean of a population. We use the name of the variable, X, with a horizontal bar over it as the symbol ("X bar") for a sample mean. Both are computed in the same way:

<math>\bar{x}={\rm arithmetic\ mean}=(x_1+\cdots+x_n)/n.<math>

The arithmetic mean is greatly influenced by outliers. For instance, reporting the "average" annual income in Redmond, Washington as the arithmetic mean of all annual incomes would yield a surprisingly high number because of Bill Gates. These distortions occur when the mean is different from the median, and the median is a superior alternative when that happens.

In certain situations, the arithmetic mean is the wrong concept of "average" altogether. For example, if a stock rose 10% in the first year, 30% in the second year and fell 10% in the third year, then it would be incorrect to report its "average" increase per year over this three year period as the arithmetic mean (10% + 30% + (−10%))/3 = 10%; the correct average in this case is the geometric mean which yields an average increase per year of only 8.8%.

If X is a random variable, then the expected value of X can be seen as the long-term arithmetic mean that occurs on repeated measurements of X. This is the content of the law of large numbers. As a result, the sample mean is used to estimate unknown expected values.

Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and the weighted mean.


Alternate notations

The arithmetic mean may also be expressed using the sum notation:

<math>\bar{x} = \frac1n\sum_{i=1}^n x_i.<math>


When used as a noun, the word arithmetic is pronounced with the emphasis on the second syllable (aRITHmetic), but when used as an adjective, as in the title of this article, the emphasis is on the third syllable (arithMETic).

See also

mean, average, summary statistics, variance, central tendency, standard deviation, inequality of arithmetic and geometric means, Muirhead's inequality

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