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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, ..., x_n}. 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:

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 numbersIn a statistical context, laws of large numbers imply that the average of a random sample from a large population is likely to be close to the mean of the whole population. In probability theory, several laws of large numbers say that the average of a seq. As a result, the sample mean is used to estimate unknown expected values.

Note that several other "means" have been defined, including the generalized meanA generalized mean or power mean is an abstraction of the arithmetic, geometric and harmonic means. If t is a non-zero real number, we can define the generalized mean with exponent t of the positive real numbers a . a as : The case t 1 yields the arithmet, the generalized f-meanIn mathematics and statistics, the generalised f- mean is the natural generalisation of the more familar means such as the arithmetic mean and the geometric mean, using a function f(x). If f is a function which maps a connected subset S of the real line t, the harmonic meanIn mathematics, the harmonic mean is one of several methods of calculating an average. The harmonic mean of the positive real numbers a . a is defined to be : The harmonic mean is never larger than the geometric mean or the arithmetic mean (see generalize, the arithmetic-geometric meanIn mathematics, the arithmetic-geometric mean M x y of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a i. a x + y / 2. We then form the geometric mean of x and y and call it g i. g is the, and the weighted meanIn statistics, given a set of data, X { x x . x and corresponding weights, W { w w . w the weighted mean is calculated as : Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. While weighted means generally behave.