variancetomean

Variancetomean, commonly referred to as the variance-to-mean ratio (VMR) or dispersion index, is a summary statistic used to describe the dispersion of a nonnegative random variable relative to its mean. If X is a random variable with mean μ = E[X] and variance σ^2 = Var(X), the variancetomean is defined as VM = σ^2 / μ, provided μ > 0. When μ = 0, the ratio is undefined.

Interpretation and properties include that VM = 1 corresponds to the dispersion of a Poisson distribution, where variance equals the mean. Values greater than 1 indicate overdispersion, meaning greater variability than a Poisson process; values less than 1 indicate underdispersion. The statistic is scale-dependent: if counts are multiplied by a constant c > 0, the VM is also multiplied by c. This means cross-dataset comparisons require careful consideration of measurement units or alternative standardizations.

Estimation from data is typically done with the sample variant s^2 / x̄, where s^2 is the sample variance and x̄ is the sample mean. Cautions include small sample sizes, zero or near-zero means, and skewed distributions, all of which can affect reliability. In practice, the variancetomean is used to assess Poisson assumptions in count data and to guide model choice; for example, overdispersion may lead to adopting a negative binomial model rather than Poisson. Related concepts include the Fano factor (often denoting Var/Mean in neuroscience and physics) and Mandel’s Q parameter, which expresses deviation from Poisson behavior as Q = VM − 1.

Applications span biology, ecology, neuroscience, reliability engineering, and quality control, where understanding dispersion relative to the mean informs data modeling and interpretation.