A CUSUM chart for monitoring a proportion when inspecting continuously

A control chart is considered for the problem of monitoring a process when all items from the process are inspected and classified into one of two categories. The objective is to detect changes in the proportion, p, of items in the first category. The control chart being considered is a cumulative sum (CUSUM) chart based on the Bernoulli observations corresponding to the inspection of the individual items. Bernoulli CUSUM charts can be constructed to detect increases in p, decreases in p, or both increases and decreases in p. The properties of the Bernoulli CUSUM chart are evaluated using exact Markov chain methods and by using a corrected diffusion theory approximation. The corrected diffusion theory approximation provides a relatively simple method of designing the chart for practical applications. It is shown that the Bernoulli CUSUM chart will detect changes in p substantially faster than the traditional approach of grouping items into samples and applying a Shewhart p-chart. The Bernoulli CUSUM chart is also better than grouping items into samples and applying a CUSUM chart to the sample statistics. The Bernoulli CUSUM chart is equivalent to a geometric CUSUM chart which is based on counting the number of items in the second category that occur between items in the first category.