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Process capability index

Statistical measure of process capability

Wikipedia contributorsAdapted for Fathomly · View authors
5 min read
Source updated December 10, 2025

The process capability index, or process capability ratio, is a statistical measure of process capability: the ability of an engineering process to produce an output within specification limits. The concept of process capability only holds meaning for processes that are in a state of statistical control. This means it cannot account for deviations which are not expected, such as misaligned, damaged, or worn equipment. Process capability indices measure how much "natural variation" a process experiences relative to its specification limits, and allows different processes to be compared to how well an organization controls them. Somewhat counterintuitively, higher index values indicate better performance, with zero indicating high deviation.

01Example for non-specialists

A company produces axles with nominal diameter 20 mm on a lathe. As no axle can be made to exactly 20.000000 mm, the designer specifies the maximum admissible deviations (called tolerances or specification limits). For instance, the requirement could be that axles need to be between 19.9 and 20.2 mm. The process capability index is a measure for how likely it is that a produced axle satisfies this requirement. The index pertains to statistical (natural) variations only. These are variations that naturally occur without a specific cause. Errors not addressed include operator errors, or play in the lathe's mechanisms resulting in a wrong or unpredictable tool position. If errors of the latter kinds occur, the process is not in a state of statistical control. When this is the case, the process capability index is meaningless.

02Introduction

If the upper and lower specification limits of the process are USL and LSL, the target process mean is T, the estimated mean of the process is {\hat {\mu }} and the estimated variability of the process (expressed as a standard deviation) is {\hat {\sigma }}, then commonly accepted process capability indices include:

Index Description
{\hat {C}}_{p}={\frac {\text{USL - LSL}}{6{\hat {\sigma }}}} Estimates what the process is capable of producing if the process mean were to be centered between the specification limits. Assumes process output is approximately normally distributed.
{\hat {C}}_{p,{\text{lower}}}={{\hat {\mu }}-{\text{LSL}} \over 3{\hat {\sigma }}} Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed.
{\hat {C}}_{p,{\text{upper}}}={{\text{USL}}-{\hat {\mu }} \over 3{\hat {\sigma }}} Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed.
{\hat {C}}_{pk}=\min {\Bigg [}{{\text{USL}}-{\hat {\mu }} \over 3{\hat {\sigma }}},{{\hat {\mu }}-{\text{LSL}} \over 3{\hat {\sigma }}}{\Bigg ]} Estimates what the process is capable of producing, considering that the process mean may not be centered between the specification limits. (If the process mean is not centered, {\hat {C}}_{p} overestimates process capability.) {\hat {C}}_{pk}<0 if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed.
{\hat {C}}_{pm}={\frac {{\hat {C}}_{p}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}} Estimates process capability around a target, T. {\hat {C}}_{pm} is always greater than zero. Assumes process output is approximately normally distributed. {\hat {C}}_{pm} is also known as the Taguchi capability index.
{\hat {C}}_{pkm}={\frac {{\hat {C}}_{pk}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}} Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed.

{\hat {\sigma }} is estimated using the sample standard deviation.

04Relationship to measures of process fallout

The mapping from process capability indices, such as Cpk, to measures of process fallout is straightforward. Process fallout quantifies how many defects a process produces and is measured by DPMO or PPM. Process yield is the complement of process fallout and is approximately equal to the area under the probability density function \Phi (\sigma )={\frac {1}{\sqrt {2\pi }}}\int _{-\sigma }^{\sigma }e^{-t^{2}/2}\,dt if the process output is approximately normally distributed.

In the short term ("short sigma"), the relationships are:

Cp Sigma level (σ) Area under the

probability density function \Phi (\sigma )

 Process yield Process fallout

(in terms of DPMO/PPM)

0.3310.682689492168.27%317311
0.6720.954499736195.45%45500
1.0030.997300203999.73%2700
1.3340.999936657599.99%63
1.6750.999999426799.9999%1
2.0060.999999998099.9999998%0.002

In the long term, processes can shift or drift significantly (most control charts are only sensitive to changes of 1.5σ or greater in process output). If there was a 1.5 sigma shift 1.5σ off of target in the processes (see Six Sigma), it would then produce these relationships:

Cp Adjusted

Sigma level (σ)

 Area under the

probability density function \Phi (\sigma )

 Process yield Process fallout

(in terms of DPMO/PPM)

0.33 1 0.3085375387 30.85% 691462
0.67 2 0.6914624613 69.15% 308538
1.00 3 0.9331927987 93.32% 66807
1.33 4 0.9937903347 99.38% 6209
1.67 5 0.9997673709 99.9767% 232.6
2.00 6 0.9999966023 99.99966% 3.40

Because processes can shift or drift significantly long term, each process would have a unique sigma shift value, thus process capability indices are less applicable as they require statistical control.

05Example

Consider a quality characteristic with target of 100.00 μm and upper and lower specification limits of 106.00 μm and 94.00 μm respectively. If, after carefully monitoring the process for a while, it appears that the process is in control and producing output predictably (as depicted in the run chart below), we can meaningfully estimate its mean and standard deviation.

If {\hat {\mu }} and {\hat {\sigma }} are estimated to be 98.94 μm and 1.03 μm, respectively, then

Index
{\hat {C}}_{p}={\frac {\text{USL - LSL}}{6{\hat {\sigma }}}}={\frac {106.00-94.00}{6\times 1.03}}=1.94
{\hat {C}}_{pk}=\min {\Bigg [}{{\text{USL}}-{\hat {\mu }} \over 3{\hat {\sigma }}},{{\hat {\mu }}-{\text{LSL}} \over 3{\hat {\sigma }}}{\Bigg ]}=\min {\Bigg [}{106.00-98.94 \over 3\times 1.03},{98.94-94 \over 3\times 1.03}{\Bigg ]}=1.60
{\hat {C}}_{pm}={\frac {{\hat {C}}_{p}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}}={\frac {1.94}{\sqrt {1+\left({\frac {98.94-100.00}{1.03}}\right)^{2}}}}=1.35
{\hat {C}}_{pkm}={\frac {{\hat {C}}_{pk}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}}={\frac {1.60}{\sqrt {1+\left({\frac {98.94-100.00}{1.03}}\right)^{2}}}}=1.11

The fact that the process is running off-center (about 1σ below its target) is reflected in the markedly different values for Cp, Cpk, Cpm, and Cpkm.

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Sources and credits

This article is adapted from the Wikipedia article “Process capability index”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Process (engineering)

In engineering, a process is a series of interrelated tasks that, together, transform inputs into a given output. These tasks may be carried out by people, nature or machines using various resources; an engineering process must be considered in the context of the agents carrying out the tasks and the resource attributes involved.

Process capability

The process capability is a measurable property of a process to the specification, expressed as a process capability index or as a process performance index (e.g., Ppk or Ppm). The output of this measurement is often illustrated by a histogram and calculations that predict how many parts will be produced out of specification (OOS).

Process performance index

In process improvement efforts, the process performance index is an estimate of the process capability of a process during its initial set-up, before it has been brought into a state of statistical control. Formally, if the upper and lower specifications of the process are USL and LSL, the estimated mean of the process is μ ^ {\displaystyle {\hat {\mu }}} , and the estimated variability of the process is σ ^ {\displaystyle {\hat {\sigma }}} , then the process performance index is defined as: P ^ p k = min [ U S L − μ ^ 3 σ ^ , μ ^ − L S L 3 σ ^ ] {\displaystyle {\hat {P}}_{pk}=\min {\Bigg [}{USL-{\hat {\mu }} \over 3{\hat {\sigma }}},{{\hat {\mu }}-LSL \over 3{\hat {\sigma }}}{\Bigg ]}} σ ^ {\displaystyle {\hat {\sigma }}} is estimated using the sample standard deviation.