Lexis ratio

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The Lexis ratio is used in statistics as a measure which seeks to evaluate differences between the statistical properties of random mechanisms where the outcome is two-valued — for example "success" or "failure", "win" or "lose". The idea is that the probability of success might vary between different sets of trials in different situations.

The measure compares the between-set variance of the sample proportions (evaluated for each set) with what the variance should be if there were no difference between in the true proprtions of success across the different sets. Thus the measure is used to evaluate how data compares to a fixed-probability-of-success Bernoulli distribution. The term "Lexis ratio" is sometimes refered to as L or Q, where

$L^2 =Q^2 = \frac{s^2}{\sigma_0^2}.$

Where $s^2 \,$ is the (weighted) sample variance derived from the observed proportions of success in sets in "Lexis trials" and $\sigma_0^2$ is the variance computed from the expected Bernoulli distribution on the basis of the overall average prooprtion of success. Trials where L falls significantly above or below 1 are known as supernormal and subnormal, respectively.