BB is correct but a little blunt. Here’s the full history in brief:
Harry confuses 1.5 Shainin Pre Control green band limits with control chart limits.
Bill Smith observes “sudden shifts” due to special causes and broadens tolerances to Cp=2.
Mikel Harry derives +/-1.5 as a “shift” in the process mean, based on tolerances in stacks of disks. This is referred to as the Z shift.
Harry realises his error and says the 1.5 “is not needed”.
Harry in about 2003 makes a new derivation of 1.5 based on errors in the estimation of sigma from sample standard deviations. For a special case of 30 points, p=.95 he multiplies Chi square factor by 3, subtracts 3 and gets “1.5”. Actual factor ranges from 1 to 18. He now calls this a “correction”, not a “shift”.
Reigle adds a new calculation he calls a “dynamic mean off-set.”: 3 / sqrt( n ) where 3 is the value for control limits and n is the subgroup size. For n=4 he gets “1.5”. Reigle says “This means that the classic Xbar chart can only detect a 1.5 sigma shift (or larger) in the process mean when subgroup size is 4”. Reigle is quite incorrect. Such data is available from ARL (Average Run Length) plots.
In other words, there is no 1.5. There is no 3.4. Six sigma tables are nonsense. Forget about 1.5.
Control limits are always +/-3 sigma.
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