Mean

In statistics, the mean is a measure of central tendency that summarizes a data set by a single representative value. The term most commonly refers to the arithmetic mean, but several related concepts exist, including the geometric mean and the harmonic mean, each with different interpretations and applicability.

The arithmetic mean of a finite set of numbers x1, x2, ..., xn is their sum divided by n: x̄ = (x1 + x2 + ... + xn) / n. For a population, the mean is denoted μ and defined as μ = (1/N) ∑ Xi; for a sample, the estimate is x̄. The sample mean is an unbiased estimator of the population mean and is linear: E[x̄] = μ.

The variance of the sample mean is Var(x̄) = σ² / n for independent, identically distributed observations with variance σ². By the law of large numbers, x̄ converges to μ as n increases, making the mean a reliable summary for sufficiently large samples.

Weighted means account for unequal contributions of observations. The weighted arithmetic mean is x̄_w = (∑ w_i x_i) / (∑ w_i), where w_i are nonnegative weights.

Geometric and harmonic means provide alternative summaries. The geometric mean is GM = (∏ x_i)^{1/n} and is appropriate for rates or multiplicative processes. The harmonic mean is HM = n / ∑ (1/x_i) and is useful for averaging rates or ratios when values are defined as time per unit.

Considerations: means are sensitive to extreme values and require data on an interval or ratio scale. They are not suitable for nominal data, and outliers can distort the arithmetic mean, while geometric and harmonic means have different sensitivities and requirements (e.g., positivity for the geometric mean).