Populationsparameters

Populationsparameters are fixed, unknown numerical characteristics that summarize a population. In statistics they are considered constants of the population, not random variables. Because a population is often too large or inaccessible, these parameters are not directly observable and must be inferred from samples.

Common examples include the population mean (μ), the population variance (σ^2), the population proportion (p), and population correlation (ρ). In a regression setting, the population regression coefficients (β) describe the relationship between variables, and in time-series the population autocorrelations may be of interest. By contrast, the corresponding quantities computed from a particular sample are sample statistics and may vary from one sample to another.

Estimation of population parameters uses observed data. Estimators are functions of the sample data that provide point estimates of the parameters. Desirable properties include unbiasedness (expectation equals the parameter), consistency (estimates converge to the parameter with larger samples), and efficiency (minimizing variance among unbiased estimators). Common estimation methods are maximum likelihood estimation and the method of moments; Bayesian methods incorporate prior information to form a posterior distribution.

Because estimates are uncertain, inference focuses on quantifying this uncertainty. Frequentist approaches yield confidence intervals, while Bayesian approaches yield credible intervals. Large-sample theory, via the central limit theorem, often allows normal approximations for many estimators.

In practice, knowledge of population parameters guides study design, hypothesis testing, and sample-size planning. In finite populations, sampling without replacement introduces a finite population correction that can affect variance estimates.