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 (cardinality). (The word set is used perhaps somewhat loosely; for example, the number 3.8 could occur more than once in such a "set".) If one particular number occurs more times than others in the set, it is called a mode. 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.

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

The symbol &mu; (Greek: mu) is used to denote the arithmetic mean of a population.

If we denote a set of data by X = { x1, x2, ..., xn}, then the sample mean is typically denoted with a horizontal bar over the variable (x̅, generally enunciated "x bar").

In practice, the difference between &mu; and x̅ is that &mu; is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may treat x̅, but not &mu;, as a random variable, attributing a probability distribution to it.

Both are computed in the same way:


 * $$\mathrm{mean} = (x_1+\cdots+x_n)/n.$$

The arithmetic mean is greatly influenced by outliers. For instance, reporting the "average" net worth in Redmond, Washington as the arithmetic mean of all annual net worths 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% + (&minus;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:


 * $$\bar{x} = \frac1n\sum_{i=1}^n x_i.$$