First of all: I'm not a statistician, so my answer may be wrong...

Basically, your statement is correct, but the question is: What is the value of the standard deviation? Normally for such such control charts, a small sample is pulled from the entire population repeatedly over time. You can calculate the standard deviation of that small sample, but must use statistical arguments to correct this to obtain the "true" standard deviation of the entire population. Or only the range of the values in the sample is taken, like in the OP's text, and again correction factors must be applied to estimate the standard deviation. These correction factors are tabulated in statistics texts for various sample sizes.

Therefore, I think that the factor 3.14 (if it's correct at all) is not pi, but one of these correction factors.