Is Your Cpk Really Above 1.33?

A supplier sends a capability study: Cpk = 1.40, thirty parts. The customer's threshold is 1.33, so it passes. But 1.40 is a point estimate from a sample, and the criterion is a statement about the process. With thirty readings, the 95% lower bound on that Cpk is about 1.02. The data are consistent with a process that will not meet the requirement.

Stratum 1.5 puts an interval on every capability index. Here is what those intervals show, how they are built, and how many parts it takes before 1.33 is a claim you can defend.

The baseline: 125 in-control rings

Keep the 25 Phase-I batches (in the Organize panel: Keep rows where Phase is one of I), then open Analyze ▸ Process Capability with Measurement = Diameter, Subgroup = Sample, and the limits entered under Spec Limits. With 1.5 the Capability Indices table splits each column into Estimate, Lower 95% and Upper 95%. Intervals are on by default.

Cpk is 1.65, interval 1.41 to 1.88. The lower bound clears 1.33 with room to spare, so this process is capable and the data say so. The level is a choice: the Statistics button holds Confidence Intervals, Confidence Level (90%, 95% or 99%) and 1.5σ Shift.

The same process, thirty parts

Now keep only the first six batches (Sample is below 6.5): thirty readings from the same stable process, against the same spec.

Cpk reads 1.33, exactly on the line, with an interval of 0.94 to 1.72. The full baseline's 1.65 sits comfortably inside that interval, and so does 1.0. Thirty parts cannot tell those two processes apart.

This is not peculiar to these rings. The table below gives the lower 95% bound for an estimated Cpk of 1.40 at different sample sizes. These figures were computed independently, with the same interval Stratum uses:

For the bound to clear 1.33 at an estimate of 1.40, you need about 860 readings. An estimate of 1.50 gets there at about 170 and 1.60 at about 75. With only thirty parts, the estimate has to be about 1.81. Subgrouped data lose a little: the pooled within-sigma has Σ(n_i−1) degrees of freedom, not N−1.

How the interval is built

Cp and Pp are a spec width over a sigma, so their interval comes straight from νσ̂²/σ² ~ χ²_ν, the textbook chi-square interval.

CPL and CPU are distances to one limit in units of 3σ̂. Because x̄ is independent of σ̂, 3√N·CPÛ follows a noncentral t with ν degrees of freedom and noncentrality 3√N·CPU. Stratum inverts that distribution exactly at each tail. Cpk and Ppk take the interval of whichever limit is nearer. That is exact for a one-sided spec. When the mean sits near the middle of the tolerance and CPL ≈ CPU, the true distribution is a minimum of two. In our simulations at n = 30, the lower bound then covers a little more than nominal (98.6% against 97.5%), so it errs on the safe side for the question “is it at least 1.33?”

Minitab documents Bissell's (1990) normal approximation for the Cpk interval, and R's qcc uses the same approximation. On every interval in this post, Bissell and the exact interval agree to within 0.004. The method is not what is at stake here: sample size is.

Two conventions to know. The interval uses the same sigma as the index, unbiasing included: by default that is pooled S_p/c₄ within subgroups, and the plain sample SD overall. It is also two-sided, so “Lower 95%” is a one-sided 97.5% bound. If your requirement is written as a one-sided 95% bound, set Confidence Level to 90%. For the Phase-I rings that lower bound is 1.45.

The false pass, caught

In In Control and Capable, running capability on all 40 batches, drift included, gave a Ppk of 1.35: a nominal pass on a process the control chart had already flagged. The interval says the same thing in capability terms.

Ppk 1.35, interval 1.21 to 1.50. Even with 200 readings, a claim of 1.33 is not supported. The interval does not replace the control chart, though. Neither a capability index nor its interval means anything until the process is stable.

Normality and the 1.5σ shift

Every index, interval and expected PPM on the Normal report assumes normal readings. Stratum runs Anderson–Darling (the Stephens A*² form) on the readings. When it rejects at 5%, a note under the indices table gives the statistic and p-value and points you to a fitted Distribution. All three subsets here pass (Phase I: A*² = 0.19, p > 0.10), so no note appears. A non-normal fit switches the intervals off rather than presenting normal-theory bounds for it.

If your customer reports in Six Sigma terms, tick 1.5σ Shift. The indices table gains Z.LT (1.5σ shift), the within Z.bench minus 1.5. The PPM table gains DPMO (1.5σ shift), the defect rate at that Z, so six sigma short-term reads 3.4.

For Phase I, the within Z.bench is 4.89, so Z.LT is 3.39 and DPMO is 351. The overall Z.bench one row up, 4.80, is the long-term figure actually measured. Over these 25 batches the process drifted far less than the convention assumes, and the table shows both numbers side by side.

Frequently asked questions

How many parts do I need to show Cpk ≥ 1.33?

It depends on your estimate. At 1.40, about 860 readings; at 1.50, about 170; at 1.60, about 75. From 30 readings, the estimate has to be about 1.81.

Is the Lower 95% bound one-sided?

No. It is the lower end of a two-sided 95% interval, which makes it a one-sided 97.5% bound. Set the level to 90% for a one-sided 95% bound.

Why might Stratum's Cpk interval differ from Minitab's?

Minitab documents Bissell's approximation, and Stratum uses the exact noncentral t for the nearer limit. At realistic n they agree to about the second decimal. The validation record lists the references.

Which sigma does the interval use?

The index's own. Cp and Cpk use the within sigma, with its degrees of freedom (pooled: Σ(n_i−1); moving range: N−1). Pp and Ppk use the overall SD on N−1.

Do the intervals hold for non-normal data?

No. They are normal theory. That is why the Anderson–Darling note exists, and why a fitted Distribution switches the intervals off.

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Related: Cp, Cpk, Pp, Ppk: Process Capability Explained → · In Control and Capable → · Q-Q plots →