# Quality Control Introductory Overview - Establishing Control Limits

Even though one could arbitrarily determine when to declare a process out of control (that is, outside the UCL-LCL range), it is common practice to apply statistical principles to do so. Elementary Concepts discusses the concept of the sampling distribution, and the characteristics of the normal distribution. The method for constructing the upper and lower control limits is a straightforward application of the principles described there.

Example. Suppose we want to control the mean of a variable, such as the size of piston rings. Under the assumption that the mean (and variance) of the process does not change, the successive sample means will be distributed normally around the actual mean. Moreover, without going into details regarding the derivation of this formula, we also know (because of the central limit theorem, and thus approximate normal distribution of the means; see, for example, Hoyer and Ellis, 1996) that the distribution of sample means will have a standard deviation of Sigma (the standard deviation of individual data points or measurements) over the square root of n (the sample size). It follows that approximately 95% of the sample means will fall within the limits m ± 1.96 * Sigma/Square Root(n) (refer to for a discussion of the characteristics of the normal distribution and the central limit theorem). In practice, it is common to replace the 1.96 with 3 (so that the interval will include approximately 99% of the sample means), and to define the upper and lower control limits as plus and minus 3 sigma limits, respectively. Note that the Quality Control module also allows you to construct charts for means with control limits that are based on the assumption of a general non-normal distribution for the underlying data.

General case. The general principle for establishing control limits just described applies to all control charts. After deciding on the characteristic we want to control, for example, the standard deviation, we estimate the expected variability of the respective characteristic in samples of the size we are about to take. Those estimates are then used to establish the control limits on the chart. Of course, you do not actually have to perform any computations, since Quality Control will estimate the sigma for the respective quality characteristic.