Mintaátlag

Mintaátlag, in statistics, is the arithmetic mean of a sample and serves as an estimator for the population mean μ. If a sample of size n contains observations x1, x2, ..., xn drawn from a population with mean μ, the mintaátlag is x̄ = (1/n) Σ xi.

As an estimator of μ, the mintaátlag is unbiased: E[x̄] = μ. Its variability is described by Var(x̄) = σ^2 / n for independent, identically distributed observations, where σ^2 is the population variance. Consequently, as the sample size grows, the dispersion of x̄ around μ shrinks and x̄ converges to μ in probability (law of large numbers). The central limit theorem further states that, for large n, x̄ is approximately normally distributed with mean μ and variance σ^2/n, provided σ^2 is finite.

In practice, the population variance is unknown, so σ^2 is replaced by the estimated variance s^2, giving a standard error SE(x̄) ≈ s / √n. The mintaátlag underpins many statistical procedures, including confidence intervals for μ (for example, x̄ ± t_{n-1, α/2} · s/√n) and tests about the population mean. It is the most common estimator of the mean in descriptive and inferential statistics.

The mintaátlag is sensitive to outliers and skewed data, which can distort the estimate of μ. In such cases, robust measures of central tendency, like the median, may be preferred. In finite populations and sampling without replacement, the variance of x̄ is adjusted by the finite population correction factor (N−n)/(N−1).