Noncentral Distributions

TL;DR

• Describes distributions with a nonzero mean and asymmetry (values not evenly distributed around the mean).
• Common variants are the noncentral t-distribution and the noncentral chi-squared distribution.
• Used when the mean differs from zero, for small-sample inference and for goodness-of-fit testing of non-normally distributed data.

Definition

A noncentral distribution is a statistical distribution that is not centered around its mean value. It has a nonzero mean and its values are not evenly distributed around that mean; instead, the distribution may be skewed or asymmetrical with a higher concentration of values on one side of the mean. Two main types are the noncentral t-distribution and the noncentral chi-squared distribution.

Explanation

A noncentral distribution departs from a centered (central) distribution by having its central parameter shifted away from zero, producing asymmetry or skew. The noncentral t-distribution applies when the t-distribution’s mean is not equal to zero and is used in contexts such as small-sample hypothesis testing when normality may not hold. The noncentral chi-squared distribution is the counterpart for chi-squared–based procedures and applies when the chi-squared distribution’s mean is not zero, for example in assessing model fit for non-normally distributed data.

Examples

Noncentral t-distribution

Suppose a researcher studies the average height of men in a certain population. The researcher collects a sample of 50 men and measures their heights, finding a sample mean of 5’10”. The researcher knows the population mean height is actually 6’0”, and the sample size is too small to accurately represent the population. In this case, the researcher could use the noncentral t-distribution to calculate the probability that the mean height of the sample is significantly different from the population mean height.

Noncentral chi-squared distribution

Suppose a researcher studies the behavior of a certain type of animal in a particular environment. The researcher collects a sample of 100 animals and observes their behavior, finding that the mean behavior of the sample is significantly different from the expected behavior of the population. In this case, the researcher could use the noncentral chi-squared distribution to calculate the probability that the mean behavior of the sample is significantly different from the expected behavior of the population.

Use cases

• Analyzing data sets that are not normally distributed.
• Testing goodness of fit of a model to data when the central assumption (mean = 0) does not hold.
• Assessing the probability that a sample mean is significantly different from an expected or population mean, particularly in small-sample settings.

Notes or pitfalls

• Noncentral distributions are appropriate when the relevant central parameter (mean) is nonzero; using central distributions in such cases can produce misleading inference.
• Small sample sizes may not accurately represent the population, which is a common scenario motivating use of noncentral t-distributions.

Related terms

• noncentral t-distribution
• noncentral chi-squared distribution
• t-distribution
• chi-squared distribution
• goodness of fit