Variance-to-mean ratio

In probability theory and statistics, the variance-to-mean ratio (VMR), like the coefficient of variation, is a measure of the dispersion of a probability distribution. It is defined as the ratio of the variance $$\ \sigma^2 $$ to the mean  $$\ \mu $$:


 * $$VMR = {\sigma^2 \over \mu }.$$

The Poisson distribution has equal variance and mean, giving it a VMR = 1. The geometric distribution and the negative binomial distribution have VMR > 1, while the binomial distribution has VMR < 1. See Cumulant.

The VMR is a good measure of the degree of randomness of a given phenomenon. This technique is also commonly used in currency management.

Example 1. For randomly diffusing particles (Brownian motion), the distribution of the number of particle inside a given volume is poissonian, i.e. VMR=1. Therefore, to assess if a given spatial pattern (assuming you have a way to measure it) is due purely to diffusion or if some particle-particle interaction is involved : divide the space into patches, Quadrats or Sample Units (SU), count the number of individuals in each patch or SU, and compute the VMR. VMRs significantly higher than 1 denote a clustered distribution, where random walk is not enough to smother the attractive inter-particle potential.