The Shewhart control chart is rather slow to react to a shift in the process mean. The fact is, most of the control procedures in today's standards manuals are obsolete since they were designed to minimize computation rather than have optimal statistical properties. There were no calculators and certainly no personal computers in the 1920's! The Cusum or cumulative sum procedure was proposed by Page in 1954 and has gradually been replacing Shewhart charts. It has been adopted as standard BS5703 by the British Standards Institute and surprisingly this procedure has only recently been shown to be optimal (see Moustakides (1986)).

Consider the data from Example 3.27 which has a nominal value ? ₀. Let the observed values (which are in fact the average of 5 observations) be denoted by X _n; n = 1, 2, (we drop the bar). When the process is in control the X _i have mean ? ₀ = 1000 and standard deviation 50/ ?5. Suppose we particularly wish to guard against a positive increase or shift in the mean to ? ₁ = 1020, the so-called rejection quality level. Define a reference parameter k and define the one-sided Cusum:

We sound an out-of-control alarm at the first n such that C _n > h where h is the signal level.

The reference or anchor value k is picked so...