midvariance

Midvariance, sometimes called mid-variance, is a term used in statistics to describe a dispersion measure that centers on a chosen central value rather than exclusively on the mean. It is not a single standardized statistic; rather, several definitions exist, making midvariance more of a family of related concepts than a single widely agreed measure.

A common interpretation defines midvariance with respect to a central value c, as M2(c) = E[(X − c)^2], where X is a random variable. In samples, the analogue is m2(c) = (1/n) ∑ (x_i − c)^2. The particular choice of c determines the specific form and interpretation of the midvariance.

Common choices for the central value c include:

- The midrange, c = (min(X) + max(X)) / 2, which yields a dispersion measure focusing on spread around the midpoint of the observed range.

- The mean, c = E[X] (or the sample mean), which makes the midvariance coincide with the ordinary variance when c is the mean.

- The median, c = median(X), which can yield a more robust measure of dispersion when outliers are present.

Relation to other concepts: when c equals the mean, the midvariance equals the usual variance. Using alternative centers produces measures that capture spread around different central tendencies and can be useful in contexts where the mean is not the most representative center.

Applications and limitations: midvariance is used in some statistical discussions and teaching as an alternative viewpoint on spread, but it is not as standardized as variance or semi-variance. Its interpretation depends on the chosen center, and there is no single accepted method for its computation in all disciplines. See also variance, semi-variance, midrange, and robust dispersion measures.