Overdispersion and underdispersion

Overdispersion and underdispersion are concepts used in statistical analysis to describe the variability of data compared to what would be expected under a particular probability distribution.

Overdispersion

Overdispersion occurs when the observed variability in the data is greater than what would be expected based on a theoretical probability distribution. In other words, the data exhibit more variation than predicted by the assumed distribution. This could happen, for example, if there are additional sources of variation or if the data are more spread out than the distribution assumes. In the context of count data, overdispersion might occur when the variance of the data exceeds the mean, which is not consistent with a Poisson or binomial distribution.

Underdispersion

Conversely, underdispersion occurs when the observed variability in the data is less than what would be expected based on the assumed distribution. In this case, the data exhibit less variation than predicted by the theoretical distribution. Underdispersion might occur if there are factors that constrain variability or if the data are more clustered or concentrated than expected.

Laney charts

Additionally, in the context of attribute control charts, Laney charts provide a means to account for overdispersion or underdispersion. For instance, a Laney P’ Chart or a Laney U’ Chart can better distinguish between common-cause and special-cause variations compared to traditional attributes charts like a P Chart or a U Chart when overdispersion or underdispersion is present. These Laney charts incorporate adjustments, such as Sigma Z, to accommodate for overdispersion or underdispersion, ensuring more accurate control limits for process monitoring.