Capability Statistics

Q: What is the difference between Cpk and Ppk?

Cpk uses the within-subgroup (short-term) standard deviation and shows the potential of the process when it is stable. Ppk uses the overall standard deviation and shows what the process actually delivered, including drift between subgroups.

Q: What is a good Cpk?

Many industries use 1.33 as a minimum for an established process and 1.67 for critical characteristics. A Cpk of 2.0 corresponds to Six Sigma performance when centered. Check your customer requirement.

Q: Can Cpk be negative?

Yes. A negative Cpk means the process mean is outside a specification limit, so more than half of the output is out of spec.

Q: Why is my Cp high but my Cpk low?

The process spread fits within the specification, but the mean is off center, so one tail crosses a limit. Re-center the process.

Q: How many data points do I need for a capability study?

At least 100 observations is a common recommendation, and more for critical features. With fewer than about 50, the interval for the index is very wide. Always report the interval.

Q: What is Z bench?

It is the capability expressed as a single normal z score that corresponds to the total expected proportion out of specification. Adding 1.5 gives the “sigma level” by the long-term convention.

Question it answers: Can this process meet the specification limits?
Data needed: 50 to 100+ stable measurements and the specification limits
Key output: Cp, Cpk, Pp, Ppk, expected ppm, and an interval
Cp / Cpk: Within (short-term) sigma; potential capability
Pp / Ppk: Overall (long-term) sigma; actual performance
Excel: Formulas with MIN, NORM.DIST, and a moving-range sigma
Minitab: Stat > Quality Tools > Capability Analysis
Why it matters: It turns spread and centering into a number customers ask for

The Idea in Plain Language

Capability statistics compare what a process does (its spread) with what the customer requires (the specification limits). They compress the question “will this process produce good parts?” into one number you can compare across processes.

There are two families, and the difference is which standard deviation you use:

Index	Standard deviation	Answers	Also called
Cp, Cpk	Within (short-term): M̄R / 1.128 or R̄ / d₂	What the process could do if it stayed centered and stable	Potential capability
Pp, Ppk	Overall (long-term): the ordinary sample standard deviation	What the process actually did, including drift and shifts	Performance

Index	Formula	Meaning
Cp	(USL − LSL) / (6σ_within)	Spec width divided by process width; ignores centering
Cpk	min[(USL − μ) / 3σ, (μ − LSL) / 3σ]	Distance from the mean to the nearer limit, in units of 3 sigma
Pp, Ppk	Same, with σ_overall	Long-term performance

Why it matters. A Cp of 2 with a badly off-center mean can still make defects. Cpk catches that. And the gap between Cpk and Ppk tells you how much of the loss is drift or instability rather than natural variation.

How Big Is Big Enough?

Cpk	Sigma level (short term)	Out of spec if centered (ppm)	Typical reading
0.67	2.0	44,431	Not capable
1.00	3.0	2,700	Marginal; needs tight control
1.33	4.0	66	Common minimum for an existing process
1.67	5.0	1	Good; common for critical characteristics
2.00	6.0	0	Six Sigma level; excellent

Requirements vary: many customers ask for Cpk of at least 1.33, or 1.67 for safety-critical features. Check the requirement for your case. All of these values assume a stable and normal process; capability is meaningless for an unstable process.

Same process spread, off-center mean. Cp = 1.67 looks excellent, but Cpk = 0.89 reveals the mean is too close to the upper limit.

Worked Example: Fill Height Capability

A filler is specified at 9.40 to 10.60 mm. Fifty consecutive measurements were taken, one per sample, after the process had been shown to be in control on an I-MR chart. Normality check: Shapiro-Wilk p = 0.57, so a normal model is reasonable.

The solid curve is based on the overall standard deviation, the dashed curve on the within standard deviation. The difference reflects slow drift between points.

Mean = 10.214; overall standard deviation s = 0.1275.
Within standard deviation: average moving range M̄R = 0.1035; σ_within = 0.1035 / 1.128 = 0.0917.
Cp = (10.6 − 9.4) / (6 × 0.0917) = 2.18.
CPU = (10.6 − 10.214) / (3 × 0.0917) = 1.40; CPL = (10.214 − 9.4) / (3 × 0.0917) = 2.96. Cpk = the smaller = 1.40.
Pp = 1.20 / (6 × 0.1275) = 1.57; Ppk = min(1.01, 2.13) = 1.01.
Expected out of spec (overall): 1,218.8 ppm; within-based: 12.6 ppm. Observed: 0 below and 0 above the limits.
95% interval for Ppk (large-sample approximation): 0.79 to 1.23.

Conclusion. The within spread would give Cp = 2.18, but the mean is -0.21 mm from the middle of the specification and the process drifts a little, so Cpk = 1.40 and Ppk = 1.01. Against a 1.33 requirement, the short-term Cpk of 1.40 passes but the long-term Ppk of 1.01 does not, and the interval for Ppk (0.79 to 1.23) is wide. The gap between Cp and Cpk is a centering problem; the gap between Cpk and Ppk is drift. Re-centering the process on 10.00 would raise Cpk toward Cp.

Capability Is an Estimate

A capability index calculated from a sample is itself uncertain. With 50 points the interval for Ppk spans about ±0.22. The interval for Cp is 1.75 to 2.61. With 30 points the interval is wider still.

Observations	Approximate half-width for Cpk near 1.33
30	± 0.36
50	± 0.28
100	± 0.20
200	± 0.14
500	± 0.09

Report the interval, not just the index. A study that shows Cpk = 1.40 from 30 points may not have shown a capable process at all.

Non-Normal and Attribute Data

Situation	What to do
Skewed measurement (time, particle size)	Fit a better distribution (lognormal, Weibull) or transform with Box-Cox or Johnson; Minitab’s nonnormal capability does both
One-sided specification (an upper limit only)	Report Cpk (CPU) or Ppk only; Cp is not defined
Pass/fail data	Use binomial capability: proportion defective and the Z bench
Defects per unit	Use Poisson capability: DPU and Z bench
Unstable process	Fix stability first. Capability indices describe a stable process only

See the Non-Normal Capability, Binomial Capability, and Poisson Capability entries.

Run It in Excel and Minitab

ExcelStep by step

Mean: =AVERAGE(A2:A51) (10.214). Overall SD: =STDEV.S(A2:A51) (0.1275).
Within SD: compute moving ranges in column B (=ABS(A3-A2)), then =AVERAGE(B3:B51)/1.128 (0.0917).
Cp =(USL-LSL)/(6*sigma_w) (2.18). Cpk =MIN((USL-mean)/(3*sigma_w),(mean-LSL)/(3*sigma_w)) (1.40). Use the overall SD for Pp and Ppk.
ppm: =(NORM.DIST(LSL,mean,sd,TRUE)+1-NORM.DIST(USL,mean,sd,TRUE))*1000000.
Z bench: =NORM.S.INV(1-ppm/1000000).
Excel has no built-in capability analysis; the formulas above are the whole method.

MinitabStep by step

Stat > Quality Tools > Capability Analysis > Normal. For individual values choose Single column and a subgroup size of 1; for subgroups, use a subgroup column or size.
Enter the lower and upper specification limits. Under Options, set the target, choose a confidence interval, and choose whether to show Z.Bench and the benchmark values.
Minitab reports Cp, Cpk, Pp, and Ppk, observed and expected ppm, and a histogram with both curves.
Non-normal data: Stat > Quality Tools > Capability Analysis > Nonnormal and choose a distribution or a transformation.
Quicker: Assistant > Capability Analysis checks stability and normality, then calculates.

Minitab session window: Capability Analysis, normal (typed excerpt, simplified)

Process Capability of Fill

Process Data
LSL            9.40000
Target                *
USL           10.60000
Sample Mean   10.21360
Sample N            50
StDev(Within) 0.09173
StDev(Overall)0.12748

Potential (Within) Capability
Cp      2.18
CPL     2.96
CPU     1.40
Cpk     1.40

Overall Capability
Pp      1.57
PPL     2.13
PPU     1.01
Ppk     1.01
Cpm        *

Observed Performance           Exp. Within Performance        Exp. Overall Performance
PPM < LSL   0.00          PPM < LSL  0.00          PPM < LSL  0.00
PPM > USL   0.00          PPM > USL  12.63          PPM > USL  1218.82
PPM Total   0.00          PPM Total  12.63          PPM Total  1218.82

Reading and Reporting

Confirm stability first with a control chart. If the process is unstable, capability numbers are not meaningful.
Check the distribution with a probability plot before using a normal-based index.
Report Cpk and Ppk together with the interval, sample size, and how the data were collected.
Compare Cp with Cpk to see whether centering is the problem, and Cpk with Ppk to see whether drift is the problem.
Say which standard deviation you used, within or overall.

A sentence you can use. The process is stable and approximately normal (n = 50); Cpk = 1.40 and Ppk = 1.01 (95% interval for Ppk 0.79 to 1.23), with an expected 1,219 ppm out of specification.

Common Mistakes

Mistake	Why it misleads	Better
Computing capability on an unstable process	The indices have no stable meaning	Establish control first
Reporting Cp only	Ignores centering	Report Cpk (and Cp for potential)
Using a normal model for skewed data	Tail estimates can be off by 10 times or more	Use a better distribution or a transformation
Mixing up Cpk and Ppk	Different standard deviations, different questions	Label which one you used
Quoting a single value from a small sample	Wide uncertainty	Give the interval and sample size
Treating specification limits as targets	A centered process is better than one just inside the limits	Aim at the target and reduce variation

Try It Yourself

A characteristic has specification limits of 20 ± 3 (17 to 23). A stable process has a mean of 21 and a within standard deviation of 0.8.

Calculate Cp and Cpk.
What fraction would be out of specification?

Show the answer

Cp = 6 / (6 × 0.8) = 1.25. CPU = (23 − 21) / (3 × 0.8) = 0.83; CPL = (21 − 17) / 2.4 = 1.67. Cpk = 0.83.

Out of specification: P(X > 23) + P(X < 17) = 0.00621 + 0.0000003 = 6,210 ppm. Centering the mean at 20 would raise Cpk to 1.25.