Variance-to-mean ratio

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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.

The VMR is a good measure of the degree of randomness of a given phenomenon.

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, count the number of individual in each patch, 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.

Example 2. Microbial gene expression. RNA is made from DNA when the RNA-polymerase positions itself in front of a gene (promoter) and polymerise the messenger RNA (mRNA) from the DNA template. Ribosomes are bound to thus synthesized mRNA and translate the codons to the protein. If these processes are purely random, (assuming that all conditions for the activation of the gene are fulfilled), the distribution of the number of RNA for the given gene among bacteria should be poissonian, i.e. VMR=1. However, the variance of the protein number distribution, is amplified by the factor $b$ defined as the mean number of proteins produced per each mRNA transcript, which is shown in recent experiments.^[1]