{"history":[],"article":{"id":"733248b2-439f-4462-89b2-45e1db8868fd","title":"Process capability index","slug":"process-capability-index","content":":::summary\n:::\n\n# \nThe **[process capability](https://bliptext.com/articles/[process](https://bliptext.com/articles/process-engineering)-capability) index,** or **[process capability](https://bliptext.com/articles/[process](https://bliptext.com/articles/process-engineering)-capability) ratio,** is a statistical measure of [process capability](https://bliptext.com/articles/[process](https://bliptext.com/articles/process-engineering)-capability): the ability of an engineering [process](https://bliptext.com/articles/process-engineering) to produce an output within [specification](https://bliptext.com/articles/specification-technical-standard) limits.\nThe concept of process capability only holds meaning for processes that are in a state of [statistical control](https://bliptext.com/articles/statistical-control).\nThis means it cannot account for deviations which are not expected, such as misaligned, damaged, or worn equipment.\nProcess capability indices measure how much [\"natural variation\"](https://bliptext.com/articles/common-cause-and-special-cause) a process experiences relative to its specification limits, and allows different processes to be compared to how well an organization controls them.\nSomewhat counterintuitively, higher index values indicate better performance, with zero indicating high deviation.\n\n# Example for non-specialists\nA company produces axles with nominal diameter 20 mm on a [lathe](https://bliptext.com/articles/lathe).\nAs no axle can be made to exactly 20 mm, the designer specifies the maximum admissible deviations (called [tolerances](https://bliptext.com/articles/engineering-tolerance) or specification limits).\nFor 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.\nThe index pertains to statistical (natural) variations only.\nThese are variations that naturally occur without a specific cause.\nErrors not addressed include operator errors, or play in the lathe's mechanisms resulting in a wrong or unpredictable tool position.\nIf errors of the latter kinds occur, the process is not in a state of statistical control.\nWhen this is the case, the process capability index is meaningless.\n\n# Introduction\nIf the upper and lower [specification](https://bliptext.com/articles/specification-technical-standard) 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](https://bliptext.com/articles/standard-deviation)) is

\\hat{\\sigma}

, then commonly accepted process capability indices include:\n

\\hat{\\sigma}

is estimated using the [sample standard deviation](https://bliptext.com/articles/unbiased-estimation-of-standard-deviation).\n\n# Recommended values\nProcess capability indices are constructed to express more desirable capability with increasingly higher values.\nValues near or below zero indicate processes operating off target (

\\hat{\\mu}

far from T) or with high variation.\nFixing values for minimum \"acceptable\" process capability targets is a matter of personal opinion, and what consensus exists varies by industry, facility, and the process under consideration.\nFor example, in the automotive industry, the [Automotive Industry Action Group](https://bliptext.com/articles/automotive-industry-action-group) sets forth guidelines in the [Production Part Approval Process](https://bliptext.com/articles/production-part-approval-process), 4th edition for recommended C_pk minimum values for critical-to-quality process characteristics.\nHowever, these criteria are debatable and several processes may not be evaluated for capability just because they have not properly been assessed.\nSince the process capability is a function of the specification, the Process Capability Index is only as good as the specification.\nFor instance, if the specification came from an engineering guideline without considering the function and criticality of the part, a discussion around process capability is useless, and would have more benefits if focused on what are the real risks of having a part borderline out of specification.\nThe loss function of [Taguchi](https://bliptext.com/articles/taguchi-methods) better illustrates this concept.\nAt least one academic expert recommends the following:\nHowever where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive.\n\n# Relationship to measures of process fallout\nThe mapping from process capability indices, such as C_pk, to measures of process fallout is straightforward.\nProcess fallout quantifies how many defects a process produces and is measured by [DPMO](https://bliptext.com/articles/defects-per-million-opportunities) or [PPM](https://bliptext.com/articles/parts-per-million).\nProcess yield is the complement of process fallout and is approximately equal to the area under the [probability density function](https://bliptext.com/articles/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](https://bliptext.com/articles/normally-distributed).\nIn the short term (\"short sigma\"), the relationships are:\nIn the long term, processes can shift or drift significantly (most [control charts](https://bliptext.com/articles/control-chart) are only sensitive to changes of 1.5σ or greater in process output).\nIf there was a 1.5 sigma shift [1.5σ](https://bliptext.com/articles/1-5) off of target in the processes (see [Six Sigma](https://bliptext.com/articles/six-sigma)), it would then produce these relationships:\nBecause 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](https://bliptext.com/articles/statistical-control).\n\n# Example\nConsider a quality characteristic with target of 100.00 [μm](https://bliptext.com/articles/micrometre) and upper and lower specification limits of 106.00 [μm](https://bliptext.com/articles/micrometre) and 94.00 [μm](https://bliptext.com/articles/micrometre) respectively.\nIf, 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](https://bliptext.com/articles/run-chart) below), we can meaningfully estimate its mean and standard deviation.\nIf

\\hat{\\mu}

and

\\hat{\\sigma}

are estimated to be 98.94 μm and 1.03 μm, respectively, then\nThe fact that the process is running off-center (about 1σ below its target) is reflected in the markedly different values for C_p, C_pk, C_pm, and C_pkm.\n# [process capability index, | upload.wikimedia.org/wikipedia/commons/d/df/ProcessCapabilityExample.svg]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T06:57:07.789Z","updated_at":"2025-01-26T06:57:07.789Z","search_vector":"'capabl':2 'index':3 'process':1","search_content":" \n \n\n# \nThe **[[process](https://bliptext.com/articles/process-engineering) capability](https://bliptext.com/articles/[process](https://bliptext.com/articles/process-engineering)-capability) index,** or **[[process](https://bliptext.com/articles/process-engineering) capability](https://bliptext.com/articles/[process](https://bliptext.com/articles/process-engineering)-capability) ratio,** is a statistical measure of [[process](https://bliptext.com/articles/process-engineering) capability](https://bliptext.com/articles/[process](https://bliptext.com/articles/process-engineering)-capability): the ability of an engineering [process](https://bliptext.com/articles/process-engineering) to produce an output within [specification](https://bliptext.com/articles/specification-technical-standard) limits.\nThe concept of process capability only holds meaning for processes that are in a state of [statistical control](https://bliptext.com/articles/statistical-control).\nThis means it cannot account for deviations which are not expected, such as misaligned, damaged, or worn equipment.\nProcess capability indices measure how much [\"natural variation\"](https://bliptext.com/articles/common-cause-and-special-cause) a process experiences relative to its specification limits, and allows different processes to be compared to how well an organization controls them.\nSomewhat counterintuitively, higher index values indicate better performance, with zero indicating high deviation.\n\n# Example for non-specialists\nA company produces axles with nominal diameter 20 mm on a [lathe](https://bliptext.com/articles/lathe).\nAs no axle can be made to exactly 20 mm, the designer specifies the maximum admissible deviations (called [tolerances](https://bliptext.com/articles/engineering-tolerance) or specification limits).\nFor 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.\nThe index pertains to statistical (natural) variations only.\nThese are variations that naturally occur without a specific cause.\nErrors not addressed include operator errors, or play in the lathe's mechanisms resulting in a wrong or unpredictable tool position.\nIf errors of the latter kinds occur, the process is not in a state of statistical control.\nWhen this is the case, the process capability index is meaningless.\n\n# Introduction\nIf the upper and lower [specification](https://bliptext.com/articles/specification-technical-standard) 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](https://bliptext.com/articles/standard-deviation)) is \\hat{\\sigma} , then commonly accepted process capability indices include:\n \\hat{\\sigma} is estimated using the [sample standard deviation](https://bliptext.com/articles/unbiased-estimation-of-standard-deviation).\n\n# Recommended values\nProcess capability indices are constructed to express more desirable capability with increasingly higher values.\nValues near or below zero indicate processes operating off target ( \\hat{\\mu} far from T) or with high variation.\nFixing values for minimum \"acceptable\" process capability targets is a matter of personal opinion, and what consensus exists varies by industry, facility, and the process under consideration.\nFor example, in the automotive industry, the [Automotive Industry Action Group](https://bliptext.com/articles/automotive-industry-action-group) sets forth guidelines in the [Production Part Approval Process](https://bliptext.com/articles/production-part-approval-process), 4th edition for recommended C pk minimum values for critical-to-quality process characteristics.\nHowever, these criteria are debatable and several processes may not be evaluated for capability just because they have not properly been assessed.\nSince the process capability is a function of the specification, the Process Capability Index is only as good as the specification.\nFor instance, if the specification came from an engineering guideline without considering the function and criticality of the part, a discussion around process capability is useless, and would have more benefits if focused on what are the real risks of having a part borderline out of specification.\nThe loss function of [Taguchi](https://bliptext.com/articles/taguchi-methods) better illustrates this concept.\nAt least one academic expert recommends the following:\nHowever where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive.\n\n# Relationship to measures of process fallout\nThe mapping from process capability indices, such as C pk , to measures of process fallout is straightforward.\nProcess fallout quantifies how many defects a process produces and is measured by [DPMO](https://bliptext.com/articles/defects-per-million-opportunities) or [PPM](https://bliptext.com/articles/parts-per-million).\nProcess yield is the complement of process fallout and is approximately equal to the area under the [probability density function](https://bliptext.com/articles/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](https://bliptext.com/articles/normally-distributed).\nIn the short term (\"short sigma\"), the relationships are:\nIn the long term, processes can shift or drift significantly (most [control charts](https://bliptext.com/articles/control-chart) are only sensitive to changes of 1.5σ or greater in process output).\nIf there was a 1.5 sigma shift [1.5σ](https://bliptext.com/articles/1-5) off of target in the processes (see [Six Sigma](https://bliptext.com/articles/six-sigma)), it would then produce these relationships:\nBecause 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](https://bliptext.com/articles/statistical-control).\n\n# Example\nConsider a quality characteristic with target of 100.00 [μm](https://bliptext.com/articles/micrometre) and upper and lower specification limits of 106.00 [μm](https://bliptext.com/articles/micrometre) and 94.00 [μm](https://bliptext.com/articles/micrometre) respectively.\nIf, 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](https://bliptext.com/articles/run-chart) below), we can meaningfully estimate its mean and standard deviation.\nIf \\hat{\\mu} and \\hat{\\sigma} are estimated to be 98.94 μm and 1.03 μm, respectively, then\nThe fact that the process is running off-center (about 1σ below its target) is reflected in the markedly different values for C p , C pk , C pm , and C pkm .\n# [process capability index, | upload.wikimedia.org/wikipedia/commons/d/df/ProcessCapabilityExample.svg]","revisionCount":0},"hasMore":false,"nextCursor":null}

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