Overdispersion and UnderdispersionOverdispersion and UnderdispersionOverdispersion exists when there is more variation in your data than you would expect based on a binomial distribution (for defectives) or a Poisson distribution (for defects). Traditional P charts and U charts assume that your rate of defectives or defects remains constant over time. However, external noise factors, which are not special causes, normally cause some variation in the rate of defectives or defects over time.
The control limits on a traditional P chart or U chart become more narrow when your subgroups are larger. If your subgroups are large enough, overdispersion can cause points to appear to be out of control when they are not.
The relationship between subgroup size and the control limits on a traditional P chart or U chart is similar to the relationship between power and a 1-sample t-test. With larger samples, the t-test has more power to detect a difference. However, if the sample is large enough, even a very small difference that is not of interest can become significant. For example, with a sample of 1,000,000 observations, a t-test may determine that a sample mean of 50.001 is significantly different from 50, even if a difference of 0.001 has no practical implications for your process.
Underdispersion is the opposite of overdispersion. Underdispersion occurs when there is less variation in your data than you would expect based on a binomial distribution (for defectives) or a Poisson distribution (for defects). Underdispersion can occur when adjacent subgroups are correlated with each other, also known as autocorrelation. For example, as a tool wears out, the number of defects may increase. The increase in defect counts across subgroups can make the subgroups more similar than they would be by chance.
When data exhibit underdispersion, the control limits on a traditional P chart or U chart may be too wide. If the control limits are too wide, you can overlook special cause variation and mistake it for common cause variation.
Use the P Chart Diagnostic or U Chart Diagnostic to test your data for overdispersion and underdispersion. If your data exhibit overdispersion or underdispersion, a Laney P' chart or Laney U' chart may more accurately distinguish between common cause variation and special cause variation than a traditional P chart or U chart.
See [5] for more discussion of overdispersion and underdispersion.