Noncentrality Parameter: Definition

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What is a Noncentrality Parameter?

noncentrality parameter
A noncentral t-distribution with a noncentrality parameter of 1.

A noncentrality parameter (NCP) is a way to distinguish noncentral distributions, which have nonzero means, from their “central” counterparts which have zero means. In other words, if a population mean is μ0, then the NCP represents the normalized difference between μ0 and μ.

If no NCP is stated, it’s usually assumed the distribution is the default: a centralized distribution.

The noncentrality parameter, usually denoted as δ, is used in many areas of statistics, like hypothesis testing and sample size calculation [1]. In power analysis, many equations are stated in terms of the NCP [2]. The NCP is also used to find confidence limits for effect sizes, because effect sizes are linear functions of noncentrality parameters [3].

Noncentrality Parameter in Hypothesis Testing

In a hypothesis test, the noncentrality parameter describes the degree of difference between the alternate hypothesis (H1) and the null hypothesis (H2). Usually, centralized distributions used for hypothesis testing, like the normal distribution or t-distribution, aren’t called the “central”—this is assumed. If the alternate hypothesis is true, then the sampling distribution is a noncentral distribution that isn’t distributed around 0, but around some other point. There are (theoretically) an infinite number of possible “other points”; the noncentrality parameter, along with degrees of freedom, tells us which of the many possible noncentral distributions fits the alternate hypothesis.


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[1] Luh, W. & Gou, J. (2011). Developing the Noncentrality Parameter for Calculating Group Sample Sizes in Heterogeneous Analysis of Variance. Journal of Experimental Education, v79 n1 p53-63 2011.
[2] Newsom, J. (2020). Power. Retrieved November 30, 2021 from:
[3] Howell, D. Confidence Intervals on Effect Size.

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