Index of dispersion

The Index of Dispersion, also known as the dispersion index, coefficient of dispersion, or relative variance, is a measure used in statistics to quantify how spread out the values of a dataset are, relative to the mean of the dataset. It is defined as the ratio of the variance to the mean. This statistical measure is particularly useful in identifying whether a dataset follows a Poisson distribution, where the index of dispersion equals 1, indicating a random distribution. Values greater than 1 suggest over-dispersion, while values less than 1 indicate under-dispersion relative to a Poisson model.

Definition

The index of dispersion is calculated using the formula:

\[D = \frac{\sigma^2}{\mu}\]

where \(D\) is the index of dispersion, \(\sigma^2\) is the variance of the dataset, and \(\mu\) is the mean of the dataset.

Applications

The index of dispersion is widely used in various fields such as ecology, where it helps in understanding the spatial distribution of species, in quality control to assess the variability of manufacturing processes, and in health sciences for analyzing the distribution of disease incidents. It is also applied in finance to measure the volatility of asset returns relative to their average.

Interpretation

- **D = 1**: The data is randomly distributed, following a Poisson distribution. - **D > 1**: Indicates over-dispersion; the data are more spread out than a Poisson distribution would suggest. This might be due to the presence of outliers, clustering, or heterogeneity in the data. - **D < 1**: Suggests under-dispersion; the data are less spread out than expected in a Poisson distribution. This could be due to negative correlation among data points or uniformity.

See Also

• Variance
• Mean
• Poisson distribution
• Statistical dispersion