X-bar Chart Upper Control Limit

Walter Shewhart's X-bar chart puts limits three standard errors either side of the grand average, but computing that standard error directly is awkward on a shop floor. The range shortcut does it with arithmetic anyone can do: $\mathit{UCL} = \ar{\ar{x}} + A_2\ar{R}$, where $A_2$ comes from a small table indexed by subgroup size. With subgroups of five, $A_2 = 0.577$, so a grand average of 50.000 and an average range of 2.000 gives limits at 51.154 and 48.846.

Control limits are not specification limits, and confusing the two is the error that undermines the entire method. Specification limits come from the customer and apply to individual parts. Control limits come from the process itself and apply to subgroup averages, which vary less than individuals by a factor of $\sqrt{n}$. Drawing the customer's tolerance on an X-bar chart makes the process look far better than it is and guarantees that real signals go unnoticed.

The $A_2$ values fall quickly with subgroup size, 1.880 at $n=2$, 1.023 at 3, 0.729 at 4, 0.577 at 5, which is just the $\sqrt{n}$ effect showing up in a table. Beyond about five the gain flattens while the sampling cost keeps rising, and that is why subgroups of four or five became the standard almost everywhere. Shewhart also had a subtler reason for small subgroups taken close together in time: he wanted the within-subgroup variation to capture only common-cause noise, so that anything else showed up as a signal between subgroups instead of being absorbed into the limits.