A more accurate representation of the expected number of false alarms in statistical quality control

We explain deviations away from the usual estimate of the expected number of false alarms per unit time in the use of a Statistical Process Control (SPC) chart. Using applied probability we develop an exact analytic expression for the expected number of false alarms per unit time. The derivation involves determining the expected value of a modulus function of a random variable. Results depend upon the probability distribution of a key random variable: the time from when an out-of-control signal is observed (and an assignable cause is present) until the process is put back into control. For the cases of normal and Erlang distributions, the usual estimate used in the design of economic control charts is biased somewhat low. The bias becomes more important when there is an appreciable chance of the process going out of control within a single inter-sampling interval.