To add detail to Steve’s response (SPC does not require the Normal Distribution or the Central Limit Theorem to work) you might want to read the following articles by Donald Wheeler. He is a Shewhart/Deming purist and before reading Shewhart’s work you might find Wheeler’s easier to digest and understand.
“Myths About Shewhart’s Control Charts
”, SPC Tool Kit column, Quality Digest, September, 1996
“Myths about Process Behavior Charts
”, Quality Digest September, 2011 (This has more detail regarding different distributions)
“Foundations of Shewhart’s Charts
”, SPC Tool Kit column, Quality Digest, October, 1996
To Steve’s point the amount of misunderstanding regarding SPC – and statistics in general is astounding. Its persistence is distressing. While certainly the hacks that jumped on the Six Sigma bandwagon to make a quick buck spread a lot of this ****, the explosion of the internet and the growth of our desire for “instant pudding” knowledge (as Deming referred to it) as well as the utter lack of humility in trying to truly understand things has really contributed to this.
If you want a bit more detail this is a nice article regarding the debate between Shewhart and Pearson
“Shewhart and the Probability Approach
” Quality Digest, November 2015