Question

Can you explain in detail (mathematically) how the empirical rule relates to the Sigma Level (+/- 6 sigma)? I've heard that "defect rates assume a 1.5 sigma shift", please explain this; does this have anything to do with z-table? Why 6 sigma and not just use the +/-3 sigma from the (68-95-99.7)% rule?

Answer 1

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Yes. On one hand we have the empirical rule of 68%-95%-99.7% of area under curve is covered under 1 -2 and 3 deviations away from mean respectively.

But the 2nd concept of "6 Sigma" is something else, although the 6 Sigma has everything to do with the Z-table. 6 Sigma is 6 deviations ( instead of just 3 deviations from the mean) , meaning a Z of +/- 6 covers leaves out just 1 in a million area beyond it. i.e. P(|Z| > 6) < 1 /1million. This concept is generally used to determine defect rate for processes. A 6 Sigma process will have less than 1 in 1 millionth defects.