Five shafts every hour
Every hour, an operator takes the last five shafts from the lathe and measures the bearing seat, which should be 48.00 ± 0.10 mm. Five numbers are written down. After a shift there are forty values, and after a week several hundred. To see what the process is doing, those values need to be summarised in a way that keeps the important information and filters out the noise. That is what X, X̄ and X̿ do.
Three levels of the same measurement
- X is a single measured value: one shaft, one diameter.
- X̄, read “X-bar” and often written Xbar, is the arithmetic mean of a subgroup, the group of values taken together, here the five shafts measured in the same hour.
- X̿, read “X-double-bar” and often written Xdbar, is the mean of all the subgroup means.
A subgroup is also called a sample. The idea is that the parts in a subgroup are made under nearly the same conditions, so the differences within it show the short-term variation of the process, and the differences between subgroups show how the process moves over time.
How they are calculated
The mean of a subgroup of n values is the sum of the values divided by n:
X̄ = (X1 + X2 + … + Xn) / n
The grand mean of k subgroups is calculated the same way, from the subgroup means:
X̿ = (X̄1 + X̄2 + … + X̄k) / k
When all subgroups have the same size, X̿ is the same as the mean of all the individual values.
Worked example (illustrative values). One hourly subgroup gives 48.02, 47.98, 48.01, 48.04 and 47.99 mm. The sum is 240.04, so X̄ = 240.04 / 5 = 48.008 mm. If the six subgroups of a shift have means of 48.008, 47.996, 48.012, 48.004, 47.990 and 48.014, then X̿ = 288.024 / 6 = 48.004 mm.
The figure shows the effect. Single values scatter widely. The subgroup means stay much closer together, and X̿ runs through the middle as the centre line.
Why means vary less than single values
Random variation partly cancels out within a subgroup: one value a little high, another a little low. Statistically, if single values have a standard deviation σ, the means of subgroups of size n have a standard deviation of σ / √n. With n = 5, the means spread less than half as much as the single values.
This is why the X̄ chart is good at detecting shifts. A change in the process centre that would be hidden among scattered single values stands out against the narrower band of the means. It also explains why control limits for X̄ are much tighter than the tolerance: they describe the spread of means, not of parts.
Means are not the whole story
A mean can look perfect while the process is getting worse. Two values of 47.92 and 48.08 average to 48.00, but both are close to the limits. That is why the X̄ chart is almost always paired with a chart of the spread within each subgroup, usually the range R.
Some other points to keep in mind:
- Means never decide acceptance. A subgroup mean inside the tolerance can hide a single part outside it. Each part is judged on its own X against the tolerance limits.
- Subgroups must be rational. If a subgroup mixes parts from two machines or two cavities, its mean is an average of two processes and tells you about neither.
- X̿ needs enough data. A grand mean from only a few subgroups is a rough estimate. Control limits are normally based on at least 20 to 25 subgroups.
In short
X tells you about one part, X̄ about the process at one moment, and X̿ about where the process is centred over time. Use X to accept parts, X̄ to watch for shifts and X̿ as the reference both are compared with.
