Cp and Cpk Explained: Formulas and Worked Examples

What Cp and Cpk measure

A capability index compares two widths. The first is the tolerance, the distance between the lower specification limit (LSL) and the upper specification limit (USL). The second is the natural spread of the process, taken as six standard deviations (6σ). For a normal distribution, 99.73 percent of the output falls within three standard deviations either side of the mean.

Cp looks only at the spread. It tells you whether the process could fit inside the tolerance if it ran exactly in the middle. Cpk also looks at where the process sits. It measures the distance from the mean to the nearer specification limit in units of three standard deviations. A process with a good Cp can still have a poor Cpk if it runs off center.

Both indices describe one characteristic, such as a bore diameter or a tightening torque, made by a process that is running steadily. They summarize how that output sits against the drawing. They are not a verdict on any single part.

The Cp and Cpk formulas

Cp divides the tolerance width by six standard deviations: Cp = (USL − LSL) / (6σ).

Cpk is worked out for each limit separately, and the smaller value is reported. The upper index is CpU = (USL − mean) / (3σ) and the lower index is CpL = (mean − LSL) / (3σ). So Cpk = min(CpU, CpL), or written out in full, Cpk = min((USL − mean) / (3σ), (mean − LSL) / (3σ)).

Because Cpk uses the nearer limit, it can never be larger than Cp. The two are equal only when the mean sits exactly in the middle of the tolerance. If the mean falls outside a specification limit, Cpk is negative.

Some characteristics have only one limit, such as a maximum runout or a minimum pull-off force. Cp cannot be calculated for them. Report the index for the limit that exists as the Cpk: CpU for an upper limit, CpL for a lower one.

• Free process capability calculator (Cp, Cpk, Pp, Ppk)

Worked example: an off-center shaft

A shaft diameter is toleranced at 10.00 ± 0.05 mm, so LSL = 9.95 mm and USL = 10.05 mm. Parts are measured in subgroups of five from normal production. The control chart shows a stable process with a mean of 10.01 mm and an average subgroup range (R-bar) of 0.0233 mm.

• Within-subgroup standard deviation: σ = R-bar / d2 = 0.0233 / 2.326 = 0.010 mm. Here d2 = 2.326 is the control chart constant for subgroups of five.
• Cp = (10.05 − 9.95) / (6 × 0.010) = 0.10 / 0.06 = 1.67
• CpU = (10.05 − 10.01) / (3 × 0.010) = 0.04 / 0.03 = 1.33
• CpL = (10.01 − 9.95) / (3 × 0.010) = 0.06 / 0.03 = 2.00
• Cpk is the smaller of CpU and CpL, so Cpk = 1.33.

Within and overall variation: Pp and Ppk

The formulas do not say which standard deviation to use. That choice is the whole difference between the capability indices (Cp, Cpk) and the performance indices (Pp, Ppk).

Cp and Cpk use the within-subgroup standard deviation, estimated from a control chart as R-bar / d2 or S-bar / c4. It reflects the common-cause variation inside each subgroup, over a short stretch of time.

Pp and Ppk use the overall standard deviation. That is the ordinary sample standard deviation s of all the individual readings. It also picks up anything that moves the process between subgroups, such as a new lot of material, a tool change or drift over a shift.

A second example shows the difference. Take a longer run of the same shaft with the same within-subgroup spread, but where the mean moved between material lots. The overall standard deviation of all its readings is 0.0125 mm. Then Pp = 0.10 / (6 × 0.0125) = 1.33 and Ppk = (10.05 − 10.01) / (3 × 0.0125) = 0.04 / 0.0375 = 1.07. Cpk still says 1.33, because the spread inside each subgroup has not changed. Ppk says 1.07, and that is closer to what the customer receives. A gap this size usually shows on the control chart as subgroup averages outside the limits, which means the process is not stable in the sense Cpk assumes.

When a process is stable, the two standard deviations are close and so are the indices. A single column of readings with no subgroups gives the overall standard deviation directly. That is what Pp and Ppk use. For Cp and Cpk, the within standard deviation comes from the moving range between consecutive readings. It is the average moving range divided by 1.128, as on an individuals chart. Some software labels Cp and Cpk short-term and Pp and Ppk long-term. What actually differs is the standard deviation: within or overall.

What Cpk says about out-of-spec parts

If the process is stable and its output follows a normal distribution, Cpk converts to an expected fraction out of specification. The nearer limit sits 3 × Cpk standard deviations from the mean. The area of the normal curve beyond it is the expected share of parts past that limit.

The third column below is the share past the nearer limit. The fourth is the total for a centered process, where both limits are the same distance away. An off-center process with the same Cpk falls between the two. In the shaft example the upper limit is 4σ from the mean and the lower limit is 6σ, so the estimate is about 32 ppm, nearly all of it oversize.

What Cpk value is good enough

No standard sets one pass mark for every product. The numbers people quote come mostly from automotive practice and from customer requirements.

• 1.00 puts the nearer limit 3σ from the mean. A centered process just fits the tolerance, with no room for a shift.
• 1.33 puts it 4σ away. Many customers set this as the minimum for ongoing production.
• 1.67 puts it 5σ away. In AIAG's Production Part Approval Process (PPAP) manual, an initial process study of a stable process meets the acceptance criteria when its index is above 1.67. From 1.33 to 1.67 the result may be acceptable after review with the customer. Below 1.33 it does not meet them.
• 2.00 puts it 6σ away. That is the six sigma target of six standard deviations between the mean and the nearer limit.

The assumptions behind every Cpk

The arithmetic always produces a number. Whether that number means anything depends on four conditions.

• The process is stable. Capability describes a process in statistical control. If the control chart shows special causes, the mean and standard deviation are moving and no single index describes the process. Find and remove the special causes first.
• The data are close to normal. The ratios themselves need no particular distribution, but the link to ppm out of spec and to the usual thresholds assumes a normal one. Check a histogram or a normal probability plot. For data that are clearly not normal, the NIST/SEMATECH Engineering Statistics Handbook describes transforming the data or using percentile-based indices.
• There is enough data. The NIST handbook says a valid estimate needs about 50 independent values, and AIAG's PPAP manual asks for at least 100 readings from 25 or more subgroups for an initial study. Even 50 values leave real uncertainty. From 50 readings, an estimated Cpk of 1.33 has an approximate 95 percent confidence interval of about 1.05 to 1.61 by the formula NIST gives.
• The readings measure the parts well. Every reading includes measurement error, and the indices count that error as process variation.

Check the measurement system before the process

The standard deviation in a capability study is the spread of the measured values. It contains the real part-to-part variation plus the variation the measurement system adds through the gage, the person using it and the method. For independent sources the variances add, so observed σ² = process σ² + measurement σ².

Take a process whose true standard deviation is 0.008 mm, measured with a system whose own standard deviation is 0.006 mm. The observed standard deviation is √(0.008² + 0.006²) = 0.010 mm. Against a 0.10 mm tolerance the true Cp is 2.08, but the study reports 1.67. The gage accounts for the difference, not the process.

So run a Gage R&R or another measurement systems analysis (MSA) on the gage that will collect the data, and confirm the gage is calibrated, before you judge the process. If the measurement system uses a large share of the tolerance, improve it first.

• Gage R&R and MSA explained
• Gage R&R calculator
• Gage bias and linearity studies

Where Axiospec fits

Axiospec is calibration management software, not SPC software. The process capability calculator on this site is a standalone tool and needs no account.

Axiospec keeps the measurement side of a capability study. It records each gage's calibrations with as-found and as-left readings, its certificates, its due dates and its Gage R&R studies. That lets you show that the gage behind a capability study was in calibration when the data was taken.