Definitions

# 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 $mu$:

$textit\left\{VMR\right\} = \left\{sigma^2 over mu \right\}.$

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 Cumulants of particular probability distributions.

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

The VMR is a particular case of the more general Fano factor, with the window chosen to be infinity.

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.