The Rate of False Signals for Control Charts with Limits Estimated from Small Samples

Due to more frequent use of short production runs and the need for shorter setup times in order to make these runs cost-effective, there has lately been a great deal of interest in the statistical properties of X ñ control charts with control limits that are based on unknown process parameters. Typically the true process distribution can be assumed to be normal, but the process mean μ and variance σ are estimated from some number m of initial subgroups of size n each. For short production runs it would be desirable to use only a small number of subgroups to estimate μ and σ in order to get on with charting the run as soon as possible. However, as a number of authors have discussed, basing the chart on estimates of μ and σ made from a small number of subgroups may give rise to some unexpected and undesirable effects. In particular, a chart with limits estimated from only a few subgroup means will tend on average to produce a greater number of false signals, which will add to the cost of production. The statistical properties of charts with limits estimated from small samples have usually been studied from the perspective of the effects of estimation on the average (expected) run length for a chart, known as its ARL. We examine the ARL and find that it is easily misunderstood and that, even when it is unambiguously defined, it has only moderate value as a focal point for the study of control charts. We argue that the rate at which false signals occur in a chart is both a more intuitive concept and a more useful one for determining a reasonable number of subgroups to sample in order to construct control limits.