- Variance-to-mean ratio
In

probability theory andstatistics , the**variance-to-mean ratio (VMR)**, like thecoefficient of variation , is a measure of the dispersion of aprobability distribution . It is defined as the ratio of thevariance $sigma^2$ to themean $mu$::$extit\{VMR\}\; =\; \{sigma^2\; over\; mu\; \}.$

The

Poisson distribution has equal variance and mean, giving it a VMR = 1. Thegeometric distribution and thenegative binomial distribution have VMR > 1, while thebinomial 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.**ee also***

Exponential distribution

*Coefficient of variation

*Coefficient of dispersion

*Fano factor

*Harmonic mean

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