Normal Distribution (bell curve, Gaussian distribution) - a symmetrical curve with one peak.
The control chart, developed by Walter Shewhart, does not rely on a normal distribution of data.
Shewhart’s Charts and the Probability Approach
by Henry R. Neave and Donald J. Wheeler:
Thus, a major problem with the probability approach to control charts is that it is totally out of contact with reality. The assumptions used for the mathematical treatment become prohibitions which are mistakenly imposed upon practice. Restrictions such as the following are commonly encountered.
"The data have to be normally distributed." "The control chart works because of the central limit theorem—therefore you have to have subgroups of at least five observations." "The chart will not work with serially-correlated (autocorrelated) data—the observations must be independent of each other before you can use a control chart." These, and others like them, are examples of the tail wagging the dog. The assumptions selected for the convenience of the mathematician are turned into prerequisites for the use of the technique. In the case of control charts, this reversal is both inappropriate and misleading.