Noncentral

Noncentral is an adjective used in mathematics, statistics, and related disciplines to describe a quantity, distribution, or parameter that is not centered at a conventional reference value, typically zero or a central location. In probability and statistics, noncentrality arises when there is a true effect or shift away from a null hypothesis, leading to nonzero means or other deviations from a central model.

In statistical distributions, noncentral forms arise by introducing a nonzero noncentrality parameter that captures the size of the effect. The noncentral chi-squared distribution, for example, describes the sum of squared independent normal variables with nonzero means. If X follows a noncentral chi-squared distribution with k degrees of freedom and noncentrality parameter λ, then X is key in characterizing the distribution of test statistics under the alternative hypothesis and in power calculations. Related are the noncentral t-distribution and the noncentral F-distribution: the noncentral t occurs when a normally distributed variable with a nonzero mean is standardized by an independent estimate of variability, and the noncentral F arises from ratios of noncentral chi-squared variables. Each noncentral distribution has its own noncentrality parameter(s) that quantify the effect size relative to the central (null) case.

The concept of noncentrality is fundamental for understanding the behavior of statistical tests under alternatives. The noncentrality parameter typically measures how far the true parameter value is from the null value in standardized units, and larger values indicate greater power to detect an effect. While the term is most common in statistics, the broader idea of noncentrality—being offset from a central reference point—appears in other mathematical contexts as well.