@Mark_Bailey already answered your auto correlation question, but I have to ask why do you check the data for normality before using a control chart. You realize there is NO assumption of normality to use a control chart. Shewhart’s argument on pages 275-277 of Economic Control of Quality of Manufactured Product. On these pages Shewhart seems to toy briefly with the idea of using the probability approach. Paraphrasing his argument on these pages, he points out that if a process were exactly stable, i.e. did unwaveringly fit some precise mathematical model, and if we knew the details of its underlying (fixed) statistical distribution, we could then work in terms of probability limits. However, he notes that, in practice, we never know the appropriate type of statistical distribution. While statisticians usually plump, almost as if it is a foregone conclusion, for their favorite distribution, the Gaussian or normal distribution, Shewhart disposes of the use of the normal distribution on page 12 of his 1939 book, Statistical Method from the Viewpoint of Quality Control." (Shewhart's Charts and the Probability Approach, Wheeler and Neave, Ninth Annual Conference of the British Deming Association, 1996)
On pages 334–335 of Out of the Crisis Deming says:
“The calculations that show where to place the control limits on a chart have their basis in the theory of probability. It would nevertheless be wrong to attach any particular figure to the probability that a statistical signal for detection of a special cause could be wrong, or that the chart could fail to send a signal when a special cause exists. The reason is that no process, except in artificial demonstrations by use of random numbers, is steady, unwavering. It is true that some books on the statistical control of quality and many training manuals for teaching control charts show a graph of the normal curve and proportions of area thereunder. Such tables and charts are misleading and derail effective study and use of control charts.”
To ADD regarding auto-correlation:
"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.
"All models are wrong, some are useful" G.E.P. Box
