## What is a Noncentrality Parameter?

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 μ

_{0}, then the NCP represents the normalized difference between μ

_{0}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 (H_{2}). 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.

## References

Image created with Desmos.com.

[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: http://web.pdx.edu/~newsomj/uvclass/ho_power.pdf

[3] Howell, D. Confidence Intervals on Effect Size.