Index of dispersion

In probability theory and statistics, the index of dispersion,^[1] dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a probability distribution: it is a measure used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model.

It is defined as the ratio of the variance ${\displaystyle \sigma ^{2}}$ to the mean ${\displaystyle \mu }$,

${\displaystyle D={\sigma ^{2} \over \mu }.}$

It is also known as the Fano factor, though this term is sometimes reserved for windowed data (the mean and variance are computed over a subpopulation), where the index of dispersion is used in the special case where the window is infinite. Windowing data is frequently done: the VMR is frequently computed over various intervals in time or small regions in space, which may be called "windows", and the resulting statistic called the Fano factor.

It is only defined when the mean ${\displaystyle \mu }$ is non-zero, and is generally only used for positive statistics, such as count data or time between events, or where the underlying distribution is assumed to be the exponential distribution or Poisson distribution.