Insights / Statistics
The difference between Cp and Cpk, what counts as a good number, and a worked example calculating both from real press-fit force data.
Process capability indices answer a specific, statistically grounded question: given the natural variation a process actually produces, how comfortably does its output fit inside the specification limits. It's a different question from "did this batch pass," which only tells you about the units you happened to measure. Capability describes the process itself, and by extension what you should expect from units you haven't built yet.
Cp and Cpk are the two most common indices, and the difference between them is the single most useful thing to understand: Cp describes how wide the process spread is relative to the tolerance band, assuming the process is perfectly centered between the limits. Cpk describes the same spread, but measured against whichever specification limit the process is actually closest to, which is what actually matters once a process drifts off-center, as real processes do.
USL and LSL are the upper and lower specification limits, x̄ is the process mean, and σ is the process standard deviation, ideally estimated from a stable, in-control process using a large enough sample to be meaningful, not a handful of convenient data points.
Cp only uses the width of the tolerance band and the process spread, it has no idea where the process is actually centered. Cpk takes the smaller of the two distances from the mean to each spec limit, which is why Cpk is always less than or equal to Cp: centering can only hurt you relative to the ideal case Cp assumes, never help.
| Cpk | What it generally indicates |
|---|---|
| Below 1.0 | The process is producing output outside spec at a meaningful rate. Not capable as-is. |
| 1.0 to 1.33 | Marginally capable. Common minimum bar for non-critical characteristics, but tight enough that drift or a bad lot can push it out of spec. |
| 1.33 to 1.67 | A common target for critical characteristics in general manufacturing. |
| 1.67 and above | High capability, often expected for safety-critical or regulated characteristics, sometimes referenced informally as approaching "six sigma" performance. |
These bands are conventions, not physical laws, different industries and different customers set their own thresholds, and a medical device characteristic tied to patient safety often warrants a higher bar than these general defaults.
A press-fit connector process specifies an insertion force between 40 N and 80 N (USL 80, LSL 40). After collecting 30 subgroups from a stable, in-control process:
| Process mean (x̄) | 68 N |
| Standard deviation (σ) | 5.5 N |
| Cp calculation | (80 − 40) / (6 × 5.5) = 40 / 33 = 1.21 |
| Distance to USL | (80 − 68) / (3 × 5.5) = 12 / 16.5 = 0.73 |
| Distance to LSL | (68 − 40) / (3 × 5.5) = 28 / 16.5 = 1.70 |
| Cpk | min(0.73, 1.70) = 0.73 |
This is exactly the pattern that makes reporting both numbers worthwhile. Cp of 1.21 looks marginally acceptable on its own, but Cpk of 0.73 shows the process is actually running close enough to the upper limit that a meaningful fraction of parts are at real risk of exceeding it. The fix here isn't to reduce variation first, it's to recenter the process closer to 60 N, the midpoint of the spec, which alone would bring Cpk much closer to Cp's value without changing the process spread at all.
There's no single universal answer, it depends on the criticality of the characteristic and your industry or customer requirements, but 1.33 is a common general-manufacturing baseline, with 1.67 or higher often expected for safety-critical characteristics.
No. Cpk can equal Cp only when the process is perfectly centered between the spec limits; any off-center shift makes Cpk lower than Cp, never higher.
Most practical guidance points to at least 25 to 30 subgroups gathered across representative production conditions, not a single short run, since the standard deviation estimate is only as trustworthy as the data behind it.
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