Central Tendency of Sample Means

As was shown earlier (Advantages of SPC: feedback, the use of individual output measurements as the basis for process adjustment is inappropriate. How, then, should the need for adjustment (and the amount of adjustment) be determined?

We can take advantage of the behaviour of the mean values of samples, as described in the central limit theorem:

If random samples of size n are taken from any population having a mean µ and a standard deviation σ, the probability distribution of the sample means approaches a normal distribution of mean µ and standard error s = σ/√n as n becomes very large. To appreciate the significance of this, consider the pharmaceutical process example. If a large number of samples, each of nine tablets (i.e. n = 9), are weighed and the means of the samples calculated, their distribution will approximate a normal distribution of mean 505.48 mg (µ) and standard error 1.82 mg (s= σ//√n = 5.45/3) This is illustrated in this figure.

Because the sample means cluster more closely around the population mean than the individual measurements, they provide a sensitive measure of any tendency for the process mean to drift out of control.