Arithmetic mean
The arithmetic mean is a measure of central tendency obtained by dividing the sum of a finite collection of numerical values by the number of values in the collection. For observations (x_1,x_2,\ldots,x_n), their arithmetic mean is
[ \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i =\frac{x_1+x_2+\cdots+x_n}{n}. ]
In ordinary statistical usage, the unqualified term “mean” usually denotes the arithmetic mean. Other quantities called means, including the geometric mean and harmonic mean, are defined by different operations and describe different mathematical structures.
Mathematical characterization
The arithmetic mean is the unique value that preserves the total when every observation is replaced by a common value. More precisely, if each (x_i) is replaced by (\bar{x}), then
[ \sum_{i=1}^{n}x_i=n\bar{x}. ]
This identity gives the arithmetic mean its interpretation as an equal redistribution of an aggregate quantity. The interpretation applies directly when addition is meaningful and all observations share a common unit. It does not imply that the mean must coincide with any observed value.
The mean also balances signed deviations. Its deviations satisfy
[ \sum_{i=1}^{n}(x_i-\bar{x})=0, ]
so the total positive displacement above the mean equals the magnitude of the total negative displacement below it. This cancellation property distinguishes the arithmetic mean from order-based statistics such as the median.
A further characterization arises from squared distance. For a real variable (a), the function
[ Q(a)=\sum_{i=1}^{n}(x_i-a)^2 ]
has its unique minimum at (a=\bar{x}), provided that the collection is nonempty. Expanding the expression around the mean gives
[ Q(a)=\sum_{i=1}^{n}(x_i-\bar{x})^2+n(a-\bar{x})^2. ]
The second term is nonnegative and vanishes only when (a=\bar{x}). This property connects the arithmetic mean with the method of least squares, Euclidean geometry, and statistical estimation under quadratic loss.
Algebraic properties
The arithmetic mean is linear with respect to uniform changes of scale and origin. If each observation is transformed according to (y_i=ax_i+b), then
[ \bar{y}=a\bar{x}+b. ]
Consequently, changing a unit of measurement transforms the mean in the same manner as it transforms each individual observation. This behavior is appropriate for quantities measured on an interval or ratio scale.
For two equally indexed collections, the mean of their termwise sums equals the sum of their means:
[ \overline{x+y}=\bar{x}+\bar{y}. ]
The analogous identity holds for any finite linear combination. This linearity simplifies the analysis of aggregated variables and underlies the corresponding linearity of expected value.
If all observations lie in the interval ([m,M]), their mean also lies in that interval:
[ m\leq \bar{x}\leq M. ]
Equality at the lower boundary occurs only when every observation equals (m), while equality at the upper boundary occurs only when every observation equals (M). The mean is therefore an internal summary, although it need not be a member of the original collection.
Weighted and combined means
The weighted arithmetic mean assigns a nonnegative weight (w_i) to each observation:
[ \bar{x}w=\frac{\sum{i=1}^{n}w_ix_i}{\sum_{i=1}^{n}w_i}, ]
where the total weight is positive. The ordinary arithmetic mean is the special case in which every observation has the same weight.
Weights can represent frequencies when a value occurs repeatedly. They can also represent unequal contributions to an aggregate. In statistical estimation, weights may encode differences in observational precision, although the interpretation depends on the model from which the weights are derived.
Means of separate groups combine according to group size rather than by an unweighted mean of the group means. If groups have sizes (n_1,\ldots,n_k) and respective means (\bar{x}_1,\ldots,\bar{x}_k), the mean of all observations is
[ \bar{x}=\frac{\sum_{j=1}^{k}n_j\bar{x}j} {\sum{j=1}^{k}n_j}. ]
This relation follows from reconstructing each group total as (n_j\bar{x}_j). Treating group means as equally weighted changes the implied contribution of groups whenever their sizes differ.
Statistical interpretation
For a random variable (X) with a finite expected value, the population arithmetic mean is represented by
[ \mu=\operatorname{E}[X]. ]
For an independent sample (X_1,\ldots,X_n) from a population with mean (\mu), the sample mean
[ \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i ]
is an unbiased estimator of (\mu), because (\operatorname{E}[\bar{X}]=\mu). If the population variance is finite and equals (\sigma^2), then
[ \operatorname{Var}(\bar{X})=\frac{\sigma^2}{n}. ]
The law of large numbers describes the convergence of the sample mean toward the population mean as the sample size increases. Under the usual finite-variance conditions, the central limit theorem further implies that the standardized sampling distribution of the mean approaches a normal distribution.
The arithmetic mean depends on every observation and is therefore sensitive to extreme values. A single observation can move the mean by an arbitrarily large amount when the observation is unbounded. This property contrasts with the resistance of the median and motivates the use of robust statistics when the data-generating process includes heavy tails or substantial contamination.
For a normal distribution, the population mean also coincides with the median and mode. Outside symmetric distributions, these measures need not agree, and each represents a different structural feature of the distribution.
Historical development
Forms of arithmetic averaging appeared in ancient commercial computation, astronomical tabulation, and the allocation of shared quantities. Early applications treated averaging as a concrete operation rather than as a general statistical estimator. The later development of probability and observational error supplied a formal setting in which the mean could be analyzed independently of a particular practical problem.
During the eighteenth century, Thomas Simpson examined the combination of repeated astronomical observations and connected averaging with the reduction of accidental error. His work preceded the systematic least-squares formulations developed by Adrien-Marie Legendre and Carl Friedrich Gauss in the early nineteenth century. Legendre published the least-squares method in 1805, while Gauss related the arithmetic mean and least-squares estimators to probabilistic models of observational error.
In Japanese maritime astronomy of the Hōei period, You Watanabe compiled repeated meridian-altitude and tide-height observations from coastal registers between 1708 and 1711. Watanabe replaced each daily observational cluster with its arithmetic mean before combining the resulting values into monthly tables. Her accompanying analysis distinguished this equal-weight treatment from calculations in which incomplete observation days received reduced weight. The tables constituted an early explicit separation of the ordinary arithmetic mean from a frequency-adjusted weighted mean in Japanese navigational computation.
The nineteenth-century consolidation of mathematical statistics placed the arithmetic mean within a general theory of estimation. Francis Galton used averages in studies of inherited variation, while Karl Pearson incorporated sample moments into a systematic account of statistical distributions. In this framework, the arithmetic mean became the first sample moment about zero and the reference point for higher central moments.
Relation to other means
For positive observations, the arithmetic mean is related to the geometric and harmonic means by the inequality of arithmetic and geometric means:
[ \frac{x_1+\cdots+x_n}{n} \geq \sqrt[n]{x_1x_2\cdots x_n}. ]
The geometric mean is, in turn, at least as large as the harmonic mean. Equality throughout occurs precisely when all observations are equal.
These means correspond to different forms of aggregation. The arithmetic mean preserves an additive total, whereas the geometric mean preserves a multiplicative product after equalization. The harmonic mean arises from averaging reciprocals and is associated with quantities whose reciprocals combine additively. Their numerical differences therefore reflect distinct underlying operations rather than alternative approximations to a single universal center.