Lesson 9: Statistical Process Control

Lessons 7 and 8 asked how variables relate to each other. This lesson turns to a different, ongoing question: once a process has been improved, how does a team watch it in real time and know, from the data alone, whether it is behaving the way it always has or whether something has actually changed? That is the job of Statistical Process Control, SPC. This lesson covers common cause versus special cause variation, how data should be collected for a control chart, how to pick the right chart for a given kind of data, control limits versus specification limits, and the basic signals that flag a process as out of control.

Common cause vs. special cause variation

Every process varies, and SPC's first job is separating two kinds of variation. Common cause variation is the routine, expected noise built into a process as it normally runs, coming from many small, ever-present sources; it produces a stable, predictable pattern over time. Special cause variation comes from something outside that routine pattern, a specific, identifiable event such as a new inspector, a damaged tool, a supplier change, or a change in material, and it shows up as a point or pattern that does not belong with the rest of the data. A process showing only common cause variation is said to be in statistical control; treating an ordinary common-cause blip as if it were a special cause, chasing every small wiggle with a new corrective action, tends to make a stable process worse, not better.

Data collection for SPC: rational subgrouping

A control chart is only as good as the subgroups used to build it. A rational subgroup is a small set of measurements collected close together in time or in production sequence, so that the variation within a subgroup reflects only ordinary common-cause noise, while variation between subgroups is where a real process shift would show up. For a GCHS punch-list inspection process, a rational subgroup might be several homes inspected by the same crew on the same day, rather than homes inspected across different weeks and different crews lumped together, which would blur a genuine shift with the extra noise of mixing different conditions inside one subgroup.

Choosing the right control chart

The right chart depends on the type of data and how it is collected: continuous, measured data (cycle time in hours, square footage) uses one chart family, while attribute, count-based data (pass/fail, number of defects) uses another, and within each family the chart also depends on subgroup size.

Two continuous-data charts track subgroup averages alongside subgroup spread: Xbar-R pairs the average with the range, and is the standard choice for smaller subgroups, roughly 2 to 9 measurements, where the range is an efficient and simple estimate of spread. Xbar-S pairs the average with the standard deviation instead, and becomes the better choice as subgroups grow larger, commonly 10 or more, because the range starts throwing away information about spread that the standard deviation still captures once there are enough points in each subgroup for it to matter.

Among attribute charts, the distinction is proportion versus count, and constant versus variable sample size. The P chart plots a proportion defective and tolerates a sample size that changes from period to period, recalculating its control limits for each subgroup's size. The NP chart plots the raw count of defective units instead of a proportion, but only works when the sample size is fixed. The U chart plots defects per unit (distinguishing a defect, a single flaw, from a defective unit, an item with one or more defects) and, like the P chart, tolerates a variable sample size. The C chart plots the raw count of defects and, like the NP chart, requires a constant sample size.

Two further charts, CuSum (cumulative sum) and EWMA (exponentially weighted moving average), are built to detect small, sustained shifts in a process mean faster than a standard Shewhart chart like those above, because they accumulate information across many points rather than judging each point mostly on its own.

Control limits vs. specification limits

Control limits and specification limits answer two different questions and come from two different places. Control limits, typically the center line plus or minus three standard deviations of the plotted statistic, are calculated from the process's own historical data; they describe what the process actually does, the voice of the process. Specification limits come from the customer or a design requirement; they describe what the process is supposed to do, the voice of the customer. A process can be perfectly in statistical control, every point inside its control limits, while still producing output outside specification limits, if the process is not capable of meeting the requirement in the first place. Confusing the two, for example plotting spec limits on a control chart, is a common and consequential mistake.

Out-of-control signals

The most basic out-of-control signal is a single point falling beyond the upper or lower control limit, roughly three standard deviations from the center line, which by itself is already an unlikely event under common cause variation alone and is treated as evidence of a special cause. Beyond that single-point rule, most SPC references also flag non-random patterns inside the limits, such as a long run of consecutive points on one side of the center line, as signals worth investigating even without a point crossing a limit outright. Any of these signals should prompt an investigation into what specific, assignable cause changed, not an automatic adjustment to the process; adjusting a stable process in response to ordinary common-cause variation, sometimes called overcontrol or tampering, is itself a way to make the output worse.

Worked example: p-chart control limits

Over 20 weeks, GCHS inspects 50 units per week (1,000 total inspections) and finds 60 units total with at least one code violation. The center line is the overall proportion defective: p̄ = 60 / 1,000 = 0.06.

The control limits use p̄ ± 3√(p̄(1-p̄)/n), with n = 50 per week. First, p̄(1-p̄) = 0.06 × 0.94 = 0.0564. Dividing by n: 0.0564 / 50 = 0.001128. Taking the square root: √0.001128 ≈ 0.0336. Multiplying by 3: 3 × 0.0336 ≈ 0.1008.

The upper control limit is UCL = 0.06 + 0.1008 ≈ 0.161. The lower control limit works out to LCL = 0.06 - 0.1008 ≈ -0.041, which is below zero; since a proportion can never be negative, the lower control limit is floored at 0, as shown in the figure. Any week with a proportion defective above 0.161 is flagged as out of control, which is exactly what happens at week 8 in the chart above.