Means

Mean is a measure of central tendency used to summarize a data set by a single representative value. The term can refer to several related concepts, with the arithmetic mean being the most commonly used.

The arithmetic mean, or simply the average, of data x1, x2, ..., xn is x̄ = (x1 + x2 + ... + xn) / n. The population mean μ is the expected value of a random variable, while the sample mean x̄ estimates μ from observed data. When observations carry weights w1, w2, ..., wn, the weighted mean is (Σ w_i x_i) / (Σ w_i). The arithmetic mean is sensitive to outliers and may not reflect the center of skewed distributions.

The geometric mean, defined for positive numbers as G = (x1 x2 ... xn)^(1/n), describes typical multiplicative growth and ratios. It is less affected by extreme values than the arithmetic mean and is often used for rates or normalized data. The geometric mean relates to the mean of logarithms: log G equals the average of log xi.

The harmonic mean, H = n / Σ (1/x_i), also defined for positive data, is appropriate for averaging quantities like rates or speeds that combine inversely. It emphasizes smaller values and can be heavily influenced by near-zero terms.

Other means include generalized or power means M_r = ((1/n) Σ xi^r)^(1/r), with r = 1 yielding the arithmetic, r = 0 yielding the geometric mean, and r = -1 yielding the harmonic mean. The mean in probability corresponds to the expected value, and the sample mean is a common estimator for the population mean, with behavior described by the Law of Large Numbers.

In practice, the choice of mean depends on the data structure and the question of interest; the mean is informative for symmetric, non-skewed data but can be misleading for skewed distributions.