The arithmetic mean of a collection of values x₁, x₂, …, xₙ is their sum divided by their count: x̄ = (x₁ + x₂ + … + xₙ) / n. The mean is a measure of central tendency — it answers “what is the typical value?” by balancing the contributions of all observations equally.
The sample mean x̄ estimates the population mean μ. It is an unbiased estimator: on average across many samples, x̄ equals μ. The mean is sensitive to outliers — a single extreme value can shift it substantially — unlike the median, which is resistant. The mean minimizes the sum of squared deviations, connecting it to variance and the method of least squares.
Other types of mean serve different purposes: the geometric mean (ⁿ√(x₁ · x₂ · … · xₙ)) is appropriate for rates and ratios, the harmonic mean (n / (1/x₁ + 1/x₂ + … + 1/xₙ)) for averaging rates, and the weighted mean assigns different importance to different observations. In probability, the expected value E[X] = Σ xᵢ · P(xᵢ) generalizes the mean to random variables.