by Oliver Yu
While control limits are approximations of 3 standard deviations, they are not 3 standard deviations.
In thermodynamics, we talk about state variables and path variables. State variables - like internal energy (U) … "is what it is." Other variables like work (w) are path variables… "its value depends on how you got there."
Standard deviation is a "state"-like parameter… if you have a set of points, the standard deviation is the standard deviation; it does not matter the order in which the data happened.
Using the same data from our previous control charting example, we see the standard deviation is 2.9 and a mean of 295. The 3 standard deviations around the average is 286 - 303.
Control limits, on the other hand, are path-like parameters that depend on the order in which it was received, and in the case of pretty random data, the control limits are 285 - 306... which is pretty close to the 3 standard devations, but not exact.
Viewing the control chart, it's obvious there are no special cause signals and there are no patterns in the data that indicate the data is out of the ordinary.
But suppose we got the same exact measurements... except this time, we found that each observed value was equal to or higher than the previous:
The standard deviation remains the same and therefore average +/- 3 standard deviations remains the same: 286 - 303. But look at the control limits... they have tightened significantly to 292 - 298.
This is because the control limits are computed from the moving range, and is when the same data shows an ascending pattern, the control limits are able to shrink and flag special cause signals where the standard deviations are not.
Apply 3 standard deviations where they are applicable; they are not applicable when identifying special cause signals of stable processes.