Populationsmittels

Populationsmittels, commonly referred to as the population mean, is a measure of central tendency that summarizes the average value of a variable across all units in a population. In statistical notation, the population mean is denoted by μ (the expected value E[X] of the variable X).

For a finite population with N units and values x1, x2, ..., xN, the population mean is μ = (1/N) ∑ xi. The corresponding population variance is σ² = (1/N) ∑ (xi − μ)², which describes the spread around the mean.

In practice, μ is usually unknown and must be estimated from sample data. The sample mean x̄ = (1/n) ∑ xi, computed from a sample of size n, serves as an estimator of μ. If the sample is random and representative, E[x̄] = μ, making x̄ an unbiased estimator. The precision of x̄ is described by its standard error, traditionally approximated as SE(x̄) ≈ σ/√n when σ is known. When sampling without replacement from a finite population, a finite population correction factor applies: SE(x̄) ≈ (σ/√n) × sqrt((N − n)/(N − 1)).

The population mean is distinct from the sample mean, which is a statistic rather than a fixed population parameter. Applications span many fields, including demography, ecology, economics, and psychology, where μ supports comparisons, hypothesis testing, and the construction of confidence intervals for the central tendency of a population.