Coefficient of dispersion

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In statistics, the coefficient of dispersion is a measure used to quantify whether a set of observed occurrences are relatively clustered or dispersed compared to a standard statistical model. In this context, the observed dataset may consist of the times of occurrence of predefined events, such as earth-quakes in a given region over a given magnitude, or of the locations in geographical space of plants of a given species. Details of such occurrences are first converted into counts of the numbers of events or occurrences in each of a set of equal-sized time- or space-regions. The coefficient of dispersion is defined as the ratio of the variance σ² to the mean μ,

where the data from which the mean and variance are calculated are the counts of the numbers of occurrences across the set of intervals. The "intervals" here may be intervals in time or small regions in space. In the usual way, population and sample values of the measure can be defined.

The relevance of the coefficient of dispersion is that it has a value of one when the probability distribution of the number of occurrences in an interval is a Poisson distribution. Thus the measure can be used to assess whether observed data can be modeled using a Poisson process.

When the coefficient of dispersion is less than 1, a dataset is said to be "under-dispersed": this condition can relate to patterns of occurrence that are more regular than the randomness associated with a Poisson process. If the coefficient of dispersion is larger than 1, a dataset is said to be "over-dispersed": this can correspond to the existence of clusters of occurrences. In terms of the interval-counts, over-dispersion corresponds to there being more intervals with low counts and more intervals with high counts, compared to a Poisson distribution: in contrast, under-dispersion is characterised by there being more intervals having counts close to the mean count, compared to a Poisson distribution.