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Cp, Cpk, Pp and Ppk: Know How and When to

Use ThemDaniela Marzagão February 26, 2010 5

For many years industries have usedCp,Cpk,Pp andPpk as statistical measures of process quality capability. Some segments in

manufacturing have specified minimal requirements for these parameters, even for some of their key documents, such as advanced

product quality planning and ISO/TS-16949. Six Sigma, however, suggests a different evaluation of process capability by measuring

against a sigma level, also known as sigma capability.

Incorporating metrics that differ from traditional ones may lead some companies to wonder about the necessity and adaptation of these

metrics. It is important to emphasize that traditional capability studies as well as the use of sigma capability measures carry a similar

purpose. Once the process is under statistical control and showing only normal causes, it is predictable. This is when it becomes

interesting for companies to predict the current process’s probability of meeting customer specifications or requirements.

Capability StudiesTraditional capability rates are calculated when a product or service feature is measured through a quantitative continuous variable,

assuming the data follows a normal probability distribution. A normal distribution features the measurement of a mean and a standard

deviation, making it possible to estimate the probability of an incident within any data set.

The most interesting values relate to the probability of data occurring outside of customer specifications. These are data appearing below

the lower specification limit (LSL) or above the upper specification limit (USL). An ordinary mistake lies in using capability studies to dea

with categorical data, turning the data into rates or percentiles. In such cases, determining specification limits becomes complex. For

example, a billing process may generate correct or incorrect invoices. These represent categorical variables, which by definition carry an

ideal USL of 100 percent error free processing, rendering the traditional statistical measures (Cp,Cpk,Pp andPpk) inapplicable to categor

variables.

When working with continuous variables, the traditional statistical measures are quite useful, especially in manufacturing. The difference

between capability rates (Cp andCpk) and performance rates (Pp andPpk) is the method of estimating the statistical population standard

deviation. The difference between the centralized rates (Cp andPp) and unilateral rates (CpkandPpk) is the impact of the mean

decentralization over process performance estimates.

The following example details the impact that the different forms of calculating capability may have over the study results of a process. A

company manufactures a product that’s acceptable dimensions, previously specified by the customer, range from 155 mm to 157 mm. T

first 10 parts made by a machine that manufactures the product and works during one period only were collected as samples during a

period of 28 days. Evaluation data taken from these parts was used to make a Xbar-S control chart (Figure 1).

Figure 1: Xbar-S Control Chart of Evaluation Data

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This chart presents only common cause variation and as such, leads to the conclusion that the process is predictable. Calculation of

process capability presents the results in Figure 2.

Figure 2: Process Capability of Dimension

Calculating CpTheCp rate of capability is calculated from the formula:

wheres represents the standard deviation for a population taken from , withs-bar representing the mean of deviation for

each rational subgroup and c4 representing a statistical coefficient of correction.

In this case, the formula considers the quantity of variation given by standard deviation and an acceptable gap allowed by specified limit

despite the mean. The results reflect the population’s standard deviation, estimated from the mean of the standard deviations within the

subgroups as 0.413258, which generates aCp of 0.81.

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Rational SubgroupsA rational subgroup is a concept developed by Shewart while he was defining control graphics. It consists of a sample in which the

differences in the data within a subgroup are minimized and the differences between groups are maximized. This allows a clearer

identification of how the process parameters change along a time continuum. In the example above, the process used to collect the

samples allows consideration of each daily collection as a particular rational subgroup.

TheCpk capability rate is calculated by the formula:

considering the same criteria of standard deviation.

In this case, besides the variation in quantity, the process mean also affects the indicators. Because the process is not perfectly

centralized, the mean is closer to one of the limits and, as a consequence, presents a higher possibility of not reaching the process

capability targets. In the example above, specification limits are defined as 155 mm and 157 mm. The mean (155.74) is closer to one of

them than to the other, leading to aCpk factor (0.60) that is lower than theCp value (0.81). This implies that the LSL is more difficult to

achieve than the USL. Non-conformities exist at both ends of the histogram.

Estimating PpSimilar to theCp calculation, the performancePp rate is found as follows:

wheres is the standard deviation of all data.

The main difference between thePp andCp studies is that within a rational subgroup where samples are produced practically at the sam

time, the standard deviation is lower. In thePp study, variation between subgroups enhances thesvalue along the time continuum, a

process which normally creates more conservativePp estimates. The inclusion of between-group variation in the calculation ofPp makes

the result more conservative than the estimate ofCp.

With regard to centralization,Pp andCp measures have the same limitation, where neither considers process centralization (mean)

problems. However, it is worth mentioning thatCp andPp estimates are only possible when upper and lower specification limits exist. Ma

processes, especially in transactional or service areas, have only one specification limit, which makes usingCp andP

p impossible (unles

the process has a physical boundary [not a specification] on the other side). In the example above, the population’s standard deviation,

taken from the standard deviation of all data from all samples, is 0.436714 (overall), giving aPp of 0.76, which is lower than the obtained

value forCp.

Estimating PpkThe difference betweenCp andPp lies in the method for calculating s, and whether or not the existence of rational subgroups is consider

CalculatingPpk presents similarities with the calculation ofCpk. The capability rate forPpk is calculated using the formula:

Once more it becomes clear that this estimate is able to diagnose decentralization problems, aside from the quantity of process variatio

Following the tendencies detected inCpk, notice that thePp value (0.76) is higher than thePpk value (0.56), due to the fact that the rate of

discordance with the LSL is higher. Because the calculation of the standard deviation is not related to rational subgroups, the standard

deviation is higher, resulting in aPpk (0.56) lower than theCpk (0.60), which reveals a more negative performance projection.

Calculating Sigma CapabilityIn the example above, it is possible to observe the incidence of faults caused by discordance, whether to the upper or lower specificatio

limits. Although flaws caused by discordance to the LSL have a greater chance of happening, problems caused by the USL will continue

occur. When calculatingCpk andPpk, this is not considered, because rates are always calculated based on the more critical side of the

distribution.

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In order to calculate the sigma level of this process it is necessary to estimate the Z bench. This will allow the conversion of the data

distribution to a normal and standardized distribution while adding the probabilities of failure above the USL and below the LSL. The

calculation is as follows:

• Above the USL:

 

•

 Below the LSL:

Summing both kinds of flaws produces the following result:

(Figure 3)

Figure 3: Distribution Z

The calculation to achieve the sigma level is represented below:

Sigma level = Zbench+ 1.5 = 1.51695 + 1.5 = 3.1695

There is great controversy about the 1.5 deviation that is usually added to the sigma level. When a great amount of data is collected ove

long period of time, multiple sources of variability will appear. Many of these sources are not present when the projection is ranged to a

period of some weeks or months. The benefit of adding 1.5 to the sigma level is seen when assessing a database with a long historical

data view. The short-term performance is typically better as many of the variables will change over time to reflect changes in business

strategy, systems enhancements, customer requirements, etc. The addition of the 1.5 value was intentionally chosen by Motorola for this

purpose and the practice is now common throughout many sigma level studies.

Comparing the MethodsWhen calculatingCp andPp, the evaluation considers only the quantity of process variation related to the specification limit ranges. This

method, besides being applicable only in processes with upper and lower specification limits, does not provide information about proces

centralization. At this point,Cpk andPpk metrics are wider ranging because they set rates according to the most critical limit.

The difference betweenCp andPp, as well as betweenCpk andPpk, results from the method of calculating standard

deviation.Cp andCpk consider the deviation mean within rational subgroups, whilePp andPpk set the deviation based on studied data. It

worth working with more conservativePp andPpk data in case it is unclear if the sample criteria follow all the prerequisites necessary to

create a rational subgroup.

Cpk andPpk rates assess process capability based on process variation and centralization. However, here only one specification limit is

considered, different from the sigma metric. When a process has only one specification limit, or when the incidence of flaws over one of

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two specification limits is insignificant, sigma level,Cpk andPpk bring very similar results. When faced with a situation where both

specification limits are identified and both have a history of bringing restrictions to the product, calculating a sigma level gives a more

precise view of the risk of not achieving the quality desired by customers.

As seen in the examples above, traditional capability rates are only valid when using quantitative variables. In cases using categorical

variables, calculating a sigma level based on flaws, defective products or flaws per opportunity, is recommended.

Ppk vs CPK

Back when I used to work in Minitab Tech Support, customers often asked me, “What’s the difference between Cpk and

Ppk? It’s a !ood "uestion, especia##$ since man$ practitioners defau#t to usin! Cpk whi#e o%er#ookin! Ppk a#to!ether& It

#ike the '()s pop duo Wham*, where Cpk is +eor!e Michae# and Ppk is that other !u$&

Poof$ hairdos st$#ed with mousse, shou#der pads, and #e! warmers aside, #et’s start b$ definin! rationa# sub!roups and

then ep#ore the difference between Cpk and Ppk&

Rational Subgroups

- rationa# sub!roup is a !roup of measurements produced under the same set of conditions& Subgroups are meant to

represent a snapshot of $our process& Therefore, the measurements that make up a sub!roup shou#d be taken from a

simi#ar point in time& .or eamp#e, if $ou samp#e / items e%er$ hour, $our sub!roup si0e wou#d be /&

Formulas, Definitions, Etc.

The !oa# of capabi#it$ ana#$sis is to ensure that a process is capab#e of meetin! customer specifications, and we use

capabi#it$ statistics such as Cpk and Ppk to make that assessment& If we #ook at the formu#as for Cpk and Ppk for norm

1distribution2 process capabi#it$, we can see the$ are near#$ identica#3

The on#$ difference #ies in the denominator for the 4pper and 5ower statistics3 Cpk is ca#cu#ated usin! the WIT6I7

standard de%iation, whi#e Ppk uses the 89:;-55 standard de%iation& Without borin! $ou with the detai#s surroundin!

the formu#as for the standard de%iations, think of the within standard de%iation as the a%era!e of the sub!roup standa

de%iations, whi#e the o%era## standard de%iation represents the %ariation of a## the data& This means that3

Cpk:

• 8n#$ accounts for the %ariation WIT6I7 the sub!roups

• <oes not account for the shift and drift between sub!roups

• Is sometimes referred to as the potential capabi#it$ because it represents the potentia# $our process has at

producin! parts within spec, presumin! there is no %ariation between sub!roups 1i&e& o%er time2

Ppk:

• -ccounts for the 89:;-55 %ariation of a## measurements taken

• Theoretica##$ inc#udes both the %ariation within sub!roups and a#so the shift and drift between them

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• Is where $ou are at the end of the pro%erbia# da$

Examples of the Difference Beteen Cpk an! Ppk

.or i##ustration, #et's consider a !ata set where / measurements were taken e%er$ da$ for =) da$s&

Example " # Similar Cpk an! Ppk

-s the !raph on the #eft side shows, there is not a #ot of shift and drift between sub!roups compared to the %ariation

within the sub!roups themse#%es& Therefore, the within and o%era## standard de%iations are simi#ar, which means Cpk a

Ppk are simi#ar, too 1at =&=> and =&), respecti%e#$2&

Example $ # Different Cpk an! Ppk

In this eamp#e, I used the same data and sub!roup si0e, but I shifted the data around, mo%in! it into different

sub!roups& 18f course we wou#d ne%er want to mo%e data into different sub!roups in practice @ I’%e Aust done it here t

i##ustrate a point&2

Since we used the same data, the o%era## standard de%iation and Ppk did not chan!e& But that’s where the simi#arities

end&

5ook at the Cpk statistic& It’s >&, which is much better than the =&=> we !ot before& 5ookin! at the sub!roups p#ot, c

$ou te## wh$ Cpk increased? The !raph shows that the points within each sub!roup are much c#oser to!ether than befo

:ar#ier I mentioned that we can think of the within standard de%iation as the a%era!e of the sub!roup standard

de%iations& So #ess %ariabi#it$ within each sub!roup e"ua#s a sma##er within standard de%iation& -nd that !i%es us a hi!

Cpk&

%o Ppk or &ot to Ppk

-nd here is where the dan!er #ies in on#$ reportin! Cpk and for!ettin! about Ppk #ike it’s +eor!e Michae#’s #esserDknow

bandmate 1no offense to whoe%er he ma$ be2& We can see from the eamp#es abo%e that Cpk on#$ te##s us part of the

stor$, so the net time $ou eamine process capabi#it$, consider both $our Cpk and $our Ppk& -nd if the process is stab

with #itt#e %ariation o%er time, the two statistics shou#d be about the same an$wa$&

Pp

The Pp inde is used to summari0e a s$stem's performance in meetin! twoDsided

specification #imits 1upper and #ower2& 5ike Ppk, it uses actua# si!ma 1si!ma of theindi%idua#s2, and shows how the s$stem is actua##$ runnin! when compared to the

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specifications& 6owe%er, it i!nores the process a%era!e and focuses on the spread& If thes$stem is not centered within the specifications, Pp a#one ma$ be mis#eadin!&

The hi!her the Pp %a#ue&&&

&&&the sma##er the spread of the s$stem s output& Pp is a measure of spread on#$& -�process with a narrow spread 1a hi!h Pp2 ma$ not meet customer needs if it is not

centered within the specifications&

If the s$stem is centered on its tar!et %a#ue&&&

&&&Pp shou#d be used in conAunction with Ppk to account for both spread and centerin!& Ppand Ppk wi## be e"ua# when the process is centered on its tar!et %a#ue& If the$ are note"ua#, the sma##er the difference between these indices, the more centered the process is

Ppk

Ppk is an inde of process performance which te##s how we## a s$stem is meetin!�specifications& Ppk ca#cu#ations use actua# si!ma 1si!ma of the indi%idua#s2, and shows ho

the s$stem is actua##$ runnin! when compared to the specifications& This inde a#so takes

into account how we## the process is centered within the specification #imits&

If Ppk is =&)&&&

&&&the s$stem is producin! &>E of its output within specifications& The #ar!er the Ppk,the #ess the %ariation between process output and specifications&

If Ppk is between ) and =&)&&&

&&&not a## process output meets specifications&

If the s$stem is centered on its tar!et %a#ue&&&

&&&Ppk shou#d be used in conAunction with the Pp inde& If the s$stem is centered on its

tar!et %a#ue, Ppk and Pp wi## be e"ua#& If the$ are not e"ua#, the sma##er the differencebetween these indices, the more centered the process is&

Pr

The Pr performance ratio is used to summari0e the actua# spread of the s$stem compared

to the spread of the specification #imits 1upper and #ower2& The #ower the Pr %a#ue, thesma##er the output spread& Pr does not consider process centerin!&

When the Pr %a#ue is mu#tip#ied b$ =)), the resu#t shows the percent of the specifications

that are bein! used b$ the %ariation in the process& Pr is ca#cu#ated usin! the actua# si!m1si!ma of the indi%idua#s2 and is the reciproca# of Pp& In other words, Pr F =GPp&

Cp

The Cp inde is used to summari0e a s$stem's abi#it$ to meet twoDsided specification #imi

1upper and #ower2& 5ike Cpk, it uses estimated si!ma and, therefore, shows the s$stem's

potentia# to meet the specifications& 6owe%er, it i!nores the process a%era!e and focuseson the spread& If the s$stem is not centered within the specifications, Cp a#one ma$ be

mis#eadin!&

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The hi!her the Cp %a#ue&&&&&the sma##er the spread of the s$stem s output& Cp is a measure of spread on#$& -�process with a narrow spread 1a hi!h Cp2 ma$ not meet customer needs if it is notcentered within the specifications&

If the s$stem is centered on its tar!et %a#ue&&&

Cp shou#d be used in conAunction with Cpk to account for both spread and centerin!& Cpand Cpk wi## be e"ua# when the process is centered on its tar!et %a#ue& If the$ are not

e"ua#, the sma##er the difference between these indices, the more centered the process is

Cpk

Cpk is a capabi#it$ inde that te##s how we## as s$stem can meet specification #imits& Cpk

ca#cu#ations use estimated si!ma and, therefore, shows the s$stem's Hpotentia#H to meetspecifications& Since it takes the #ocation of the process a%era!e into account, the process

does not need to be centered on the tar!et %a#ue for this inde to be usefu#&

If Cpk is =&)&&&

&&&the s$stem is producin! &>E of its output within specifications& The #ar!er the Cpk,the #ess %ariation $ou wi## find between the process output and specifications&

If Cpk is between ) and =&)&&&&&&not a## process output meets specifications&

If the s$stem is centered on its tar!et %a#ue&&&

&&&Cpk shou#d be used in conAunction with the Cp inde& Cpk and Cp wi## be e"ua# when thprocess is centered on its tar!et %a#ue& If the$ are not e"ua#, the sma##er the difference

between these indices, the more centered the process is&

Cpm

The Cpm inde indicates how we## the s$stem can produce within specifications& Itsca#cu#ation is simi#ar to Cp, ecept that si!ma is ca#cu#ated usin! the tar!et %a#ue instead

of the mean& The #ar!er the Cpm, the more #ike#$ the process wi## produce output that

meets specifications and the tar!et %a#ue&

Cr

The Cr capabi#it$ ratio is used to summari0e the estimated spread of the s$stem compare

to the spread of the specification #imits 1upper and #ower2& The #ower the Cr %a#ue, thesma##er the output spread& Cr does not consider process centerin!&

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'hen the Cr (alue is multiplie! b) "**, the result shos the percent of thespecifications that are being use! b) the (ariation in the process. Cr is calculate

using an estimate! sig ma an! is the reciprocal of Cp. +n other or!s, Cr "-Cp

 Process Capability (Cp, Cpk) and Process

Performance (Pp, Ppk) – What is the Difference?In the Six Sigma quality methodology, process performance is reported to the organization as a sigma level. The higher the sigma level,

better the process is performing.

Another way to report process capability and process performance is through the statistical measurements ofCp,Cpk,Pp, andPpk. This

article will present definitions, interpretations and calculations forCpkandPpk though the use of forum quotations. Thanks to everyone be

that helped contributed to this excellent reference.

Jump To The Following Sections:

• Definitions

•

InterpretingCp,Cpk• InterpretingPp,Ppk

• Differences BetweenCpk andPpk

• CalculatingCpk andPpk

Definitions

 Cp= Process Capability. A simple and straightforward indicator of process capability.

 Cpk= Process Capability Index. Adjustment ofCp for the effect of non-centered distribution.

 Pp= Process Performance. A simple and straightforward indicator of process performance.

 Ppk= Process Performance Index. Adjustment ofPp for the effect of non-centered distribution.

InterpretingCp,Cpk

“Cpk is an index (a simple number) which measures how close a process is running to its specification limits, relative to the natural variabof the process. The larger the index, the less likely it is that any item will be outside the specs.”Neil Polhemus

“If you hunt our shoot targets with bow, darts, or gun try this analogy. If your shots are falling in the same spot forming a good group this

a highCp, and when the sighting is adjusted so this tight group of shots is landing on the bullseye, you now have a highCpk.”Tommy

“Cpk measures how close you are to your target and how consistent you are to around your average performance. A person may be

performing with minimum variation, but he can be away from his target towards one of the specification limit, which indicates lowerCpk,

whereasCp will be high. On the other hand, a person may be on average exactly at the target, but the variation in performance is high (b

still lower than the tolerance band (i.e., specification interval). In such case alsoCpk will be lower, butCp will be high.Cpk will be higher on

when you r meeting the target consistently with minimum variation.” Ajit

“You must have aCpk of 1.33 [4 sigma] or higher to satisfy most customers.” Joe Perito

“Consider a car and a garage. The garage defines the specification limits; the car defines the output of the process. If the car is only a libit smaller than the garage, you had better park it right in the middle of the garage (center of the specification) if you want to get all of the

car in the garage. If the car is wider than the garage, it does not matter if you have it centered; it will not fit. If the car is a lot smaller than

garage (Six Sigma process), it doesn’t matter if you park it exactly in the middle; it will fit and you have plenty of room on either side. If y

have a process that is in control and with little variation, you should be able to park the car easily within the garage and thus meet custo

requirements.Cpk tells you the relationship between the size of the car, the size of the garage and how far away from the middle of the

garage you parked the car.”Ben

“The value itself can be thought of as the amount the process (car) can widen before hitting the nearest spec limit (garage door edge).

Cpk=1/2 means you’ve crunched nearest the door edge (ouch!)

Cpk=1 means you’re just touching the nearest edge

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Cpk=2 means your width can grow 2 times before touching

Cpk=3 means your width can grow 3 times before touching”Larry Seibel

Interpreting Pp, Ppk

“Process Performance Index basically tries to verify if the sample that you have generated from the process is capable to meet Custome

CTQs (requirements). It differs from Process Capability in that Process Performance only applies to a specific batch of material. Sample

from the batch may need to be quite large to be representative of the variation in the batch. Process Performance is only used when

process control cannot be evaluated. An example of this is for a short pre-production run. Process Performance generally uses sample

sigma in its calculation; Process capability uses the process sigma value determined from either the Moving Range, Range or Sigma

control charts.”PraneetDifferences BetweenCpk andPpk

“Cpk is for short term,Ppk is for long term.”Sundeep Singh

“Ppk produces an index number (like 1.33) for the process variation.Cpk references the variation to your specification limits. If you just wa

to know how much variation the process exhibits, aPpk measurement is fine. If you want to know how that variation will affect the ability o

your process to meet customer requirements (CTQ’s), you should useCpk.”Michael Whaley

“It could be argued that the use ofPpk andCpk (with sufficient sample size) are far more valid estimates of long and short term capability

processes since the 1.5 sigma shift has a shaky statistical foundation.”Eoin

“Cpk tells you what the process is CAPABLE of doing in future, assuming it remains in a state of statistical control.Ppk tells you how the

process has performed in the past. You cannot use it predict the future, like withCpk, because the process is not in a state of control. The

values forCpk andPpk will converge to almost the same value when the process is in statistical control. that is because sigma and thesample standard deviation will be identical (at least as can be distinguished by an F-test). When out of control, the values will be distinct

different, perhaps by a very wide margin.” Jim Parnella

“Cp andCpk are for computing the index with respect to the subgrouping of your data (different shifts, machines, operators, etc.),

whilePp andPpk are for the whole process (no subgrouping). For bothPpk andCpk the ‘k’ stands for ‘centralizing facteur’ – it assumes the

index takes into consideration the fact that your data is maybe not centered (and hence, your index shall be smaller). It is more realistic t

usePp andPpk thanCp orCpk as the process variation cannot be tempered with by inappropriate subgrouping. However,Cp andCpk can b

very useful in order to know if, under the best conditions, the process is capable of fitting into the specs or not.It basically gives you the b

case scenario for the existing process.”Chantal

“Cp should always be greater than 2.0 for a good process which is under statistical control. For a good process under statistical

control,Cpk should be greater than 1.5.”Ranganadha Kumar“As forPpk /Cpk, they mean one or the other and you will find people confusing the definitions and you WILL find books defining them vers

and vice versa. You will have to ask the definition the person is using that you are talking to.” Joe Perito

“I just finished up a meeting with a vendor and we had a nice discussion ofCpk vs.Ppk. We had the definitions exactly reversed between

The outcome was to standardize on definitions and move forward from there. My suggestion to others is that each company have a

procedure or document (we do not), which has the definitions ofCpk andPpk in it. This provides everyone a standard to refer to for WHEN

we forget or get confused.” John Adamo

“The Six Sigma community standardized on definitions ofCp,Cpk,Pp, andPpk fromAIAG SPC manual page 80. You can get the manua

about $7.”Gary

CalculatingCpk andPpk

“Pp = (USL – LSL)/6*Std.devCpl = (Mean – LSL)/3*Std.dev

Cpu = (USL – Mean)/3*Std.dev

 Cpk= Min (Cpl,Cpu)”Ranganadha Kumar

“Cpk is calculated using an estimate of the standard deviation calculated using R-bar/d2.Ppk uses the usual form of the standard deviatio

the root of the variance or the square root of the sum of squares divided byn – 1. The R-bar/D2 estimation of the standard deviation has

smoothing effect and theCpk statistic is less sensitive to points which are further away from the mean than isPpk.”Eoin

“Cpk is calculated using RBar/d2 or SBar/c4 for Sigma in the denominator of you equation. This calculation for Sigma REQUIRES the

process to be in a state of statistical control. If not in control, your calculation of Sigma (and hence Cpk) is useless – it is only valid when

control.” Jim Parnella

“You can have a ‘good’Cpk yet still have data outside the specification, and the process needs to be in control before evaluatingCpk.”Ma

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Process capability indexFrom Wikipedia, the free encyclopedia

"Cpk" redirects here. For other uses, see CPK.

Inprocess improvement efforts, theprocess capability index orprocess capability ratio is a statistical measure

ofprocess capability: the ability of a process to produce output within specification limits.[1] The concept of process

capability only holds meaning for processes that are in a state ofstatistical control. Process capability indices measuhow much"natural variation" a process experiences relative to its specification limits and allows different processes t

be compared with respect to how well an organization controls them.

If the upper and lowerspecification limits of the process are USL and LSL, the target process mean is T, the estimate

mean of the process is and the estimated variability of the process (expressed as astandard deviation) is , then

commonly accepted process capability indices include:

Index Description

Estimates what the process is capable of producing if the process mea

were to be centered between the specification limits. Assumes process

output is approximately normally distributed.

Estimates process capability for specifications that consist of a lower

limit only (for example, strength). Assumes process output is

approximately normally distributed.

Estimates process capability for specifications that consist of an upper

limit only (for example, concentration). Assumes process output is

approximately normally distributed.

Estimates what the process is capable of producing, considering that t

process mean may not be centered between the specification limits. (If

the process mean is not centered, overestimates process

capability.) if the process mean falls outside of thespecification limits. Assumes process output is approximately normally

distributed.

Estimates process capability around a target, T. is always greate

than zero. Assumes process output is approximately normally

distributed. is also known as theTaguchi capability index.[2]

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Estimates process capability around a target, T, and accounts for an off

center process mean. Assumes process output is approximately

normally distributed.

 is estimated using thesample standard deviation.

Contents

 [hide]

1 Recommended values

2 Relationship to measures of process fallout

3 Example

4 See also

5 References

Recommended values[edit]

Process capability indices are constructed to express more desirable capability with increasingly higher values. Value

near or below zero indicate processes operating off target ( far from T) or with high variation.

Fixing values for minimum "acceptable" process capability targets is a matter of personal opinion, and what consens

exists varies by industry, facility, and the process under consideration. For example, in the automotive industry,

the Automotive Industry Action Group sets forth guidelines in theProduction Part Approval Process, 4th edition for

recommended Cpk minimum values for critical-to-quality process characteristics. However, these criteria are debatabl

and several processes may not be evaluated for capability just because they have not properly been assessed.

Since the process capability is a function of the specification, the Process Capability Index is only as good as the

specification . For 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

what are the real risks of having a part borderline out of specification. The loss function ofTaguchi better illustrates th

concept.

At least one academic expert recommends[3] the following:

SituationRecommended minimum process

capability for two-sided specifications

Recommended minimum process

capability for one-sided specification

Existing process 1.33 1.25

New process 1.50 1.45

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Safety or critical parameter

for existing process1.50 1.45

Safety or critical parameter

for new process1.67 1.60

Six Sigma quality process 2.00 2.00

It should be noted though that where a process produces a characteristic with a capability index greater than 2.5, the

unnecessary precision may be expensive.[4]

Relationship to measures of process fallout[edit]

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 orPPM. Process yield is, of

course, the complement of process fallout and is approximately equal to the area under theprobability density

function  if the process output is approximatelynormally distributed.

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

CpkSigma level

(σ)

Area under the probability density

function 

Process

yield

Process fallout (in terms of

DPMO/PPM)

0.3

31 0.6826894921 68.27% 317311

0.6

72 0.9544997361 95.45% 45500

1.0

03 0.9973002039 99.73% 2700

1.3

34 0.9999366575 99.99% 63

1.6

75 0.9999994267 99.9999% 1

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2.0

06 0.9999999980 99.9999998% 0.002

In the long term, processes can shift or drift significantly (mostcontrol charts are only sensitive to changes of 1.5σ o

greater in process output), so process capability indices are not applicable as they requirestatistical control.

Example[edit]

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 a

producing output predictably (as depicted in therun chart below), we can meaningfully estimate its mean and standa

deviation.

If and are estimated to be 98.94 μm and 1.03 μm, respectively, then

Index

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The fact that the process is running off-center (about 1σ below its target) is reflected in the markedly different values

Cp, Cpk, Cpm, and Cpkm.

Process Capability Index - Cp vs. Cpk Visual Animation

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Thoughts about Process Capability - Cp and Cpk

Process Capability is required by TS 16949 and is part of the typical APQP process. Process capability compares the outpof an incontrol process to the specification limits by usin! capability indices. The comparison is made by formin! the ratio the spread bet"een the process specifications #the specification $"idth$% to the spread of the process &alues' as measuredby 6 process standard de&iation units #the process $"idth$%.

Process Capability Indices - A process capability inde( uses both the process &ariability and the process specifications tdetermine "hether the process is $capable$ )e are often required to compare the output of a stable process "ith the procspecifications and ma*e a statement about ho" "ell the process meets specification. To do this "e compare the natural&ariability of a stable process "ith the process specification limits. A capable process is one "here almost all the

measurements fall inside the specification limits. There are se&eral statistics that can be used to measure the capability of

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process+ Cp' Cp*' Cpm.,ost capability indices estimates are &alid only if the sample si-e used is lar!e enou!h. /ar!eenou!h is !enerally thou!ht to be about 0 independent data &alues.

The Cp' Cp*' and Cpm statistics assume that the population of data &alues is normally distributed.

Capability Analysis - Process Capability Indices

Process range. 2irst' it is customary to establish the 3 5 sigma limits around the nominal specifications. Actually'the sigma limits should be the same as the ones used to brin! the process under control usin! She"hart control charts.These limits denote the range of the process #i.e.' process range%. f "e use the 3 5 sigma limits then' based on the normdistribution' "e can estimate that appro(imately 997 of all piston rin!s fall "ithin these limits.

Specification limits LSL, SL. 8sually' en!ineerin! requirements dictate a ran!e of acceptable &alues. n our e(ample' itmay ha&e been determined that acceptable &alues for the piston rin! diameters "ould be 4. 3 .: millimeters. Thus'the lower specification limit (LSL) for our process is 4. .: ; 5.9<= the upper specification limit (USL) is 4. 3 .: ;4.:. The difference bet"een 8S/ and /S/ is called the specification range. Potential capability !Cp". This is the simpleand most strai!htfor"ard indicator of process capability. t is defined as the ratio of the specification ran!e to the processran!e= usin! 3 5 sigma limits "e can e(press this inde( as+ Cp ; #8S//S/%#6>Si!ma% Put into "ords' this ratio e(pressethe proportion of the ran!e of the normal cur&e that falls "ithin the en!ineerin! specification limits #pro&ided that the mean on tar!et' that is' that the process is centered ' see belo"%. ?hote #19<<% reports that prior to the "idespread use of statisticquality control techniques #prior to 19<%' the normal quality of 8S manufacturin! processes "as appro(imately Cp ; .6. Tmeans that the t"o 55: percent tail areas of the normal cur&e fall outside specification limits. As of 19<<' only about 57 8S processes are at or belo" this le&el of quality #see ?hote' 19<<' p. 01%. deally' of course' "e "ould li*e this inde( to be!reater than 1' that is' "e "ould li*e to achie&e a process capability so that no #or almost no% items fall outside specificatiolimits. nterestin!ly' in the early 19<s the @apanese manufacturin! industry adopted as their standard Cp ; 1.55 Theprocess capability required to manufacture hi!htech products is usually e&en hi!her than this= ,inolta has established aCp inde( of :. as their minimum standard #?hote' 19<<' p. 05%' and as the standard for its suppliers. Bote that hi!h procescapability usually implies lo"er' not hi!her costs' ta*in! into account the costs due to poor quality. )e "ill return to this poshortly. Capability ratio !Cr ". This inde( is equi&alent to Cp= specifically' it is computed as 1Cp #the in&erse ofCp%. Lo#er$upper potential capability% Cpl, Cpu. A maor shortcomin! of the Cp #and Cr % inde( is that it may yield erroneouinformation if the process is not on tar!et' that is' if it is not centered . )e can e(press noncenterin! &ia the follo"in!quantities. 2irst' upper and lo"er potential capability indices can be computed to reflect the de&iation of the obser&ed procmean from the /S/ and 8S/.. Assumin! 3 5 sigmalimits as the process ran!e' "e compute+ Cpl ; #,ean /S/%5>Si!maandCpu ; #8S/ ,ean%5>Si!ma Db&iously' if these &alues are not identical to each other' then the process is not centered. &ocentering correction !'". )e can correct Cp for the effects of noncenterin!. Specifically' "e can compute+ E;abs#F ,ean%#1:>#8S/ /S/%% "here F ; #8S/3/S/%:. This correction factor e(presses the noncenterin! #tar!et specification

minus mean% relati&e to the specification ran!e. (emonstrated e)cellence !Cpk".2inally' "e can adust Cp for the effect ofnoncenterin! by computin!+ Cp* ; #1*%>Cp f the process is perfectly centered' then k  is equal to -ero' and Cp* is equal to Go"e&er' as the process drifts from the tar!et specification' k increases and Cp* becomes smaller than Cp. PotentialCapability II% Cpm. A recent modification #Chan' Chen!' H Spirin!' 19<<% to Cp is directed at adustin! the estimateof sigma for the effect of #random% noncenterin!. Specifically' "e may compute the alternati&e sigma (Sigma2 % as+ Si!ma:

 #(i  TS%:#n1%J� "here+Si!ma: is the alternati&e estimate of sigma(i  is the &alue of the i th obser&ation in the sample

TS  is the tar!et or nominal specificationn  is the number of obser&ations in the sample )e may then use this alternati&e estimate of sigma to compute Cp asbefore= ho"e&er' "e "ill refer to the resultant inde( as Cpm.

Process Performance &s. Process Capability

)hen monitorin! a process &ia a quality control chart #e.!.' the Kbar and Lchart=% it is often useful to compute the capabiindices for the process. Specifically' "hen the data set consists of multiple samples' such as data collected for the qualitycontrol chart' then one can compute t"o different indices of &ariability in the data. Dne is the re!ular standard de&iation forobser&ations' i!norin! the fact that the data consist of multiple samples= the other is to estimate the processs inherent&ariation from the "ithinsample &ariability. 2or e(ample' "hen plottin! Kbar and Lcharts one may use the commonestimator Lbard: for the process si!ma #e.!.' see Funcan' 194= ,ont!omery' 19<0' 1991%. Bote ho"e&er' that thisestimator is only &alid if the process is statistically stable. 2or a detailed discussion of the difference bet"een the totalprocess &ariation and the inherent &ariation refer to ASQCAAM reference manual #ASQCAAM' 1991' pa!e <%.

)hen the total process &ariability is used in the standard capability computations' the resultin! indices are usually referredas process performance indices #as they describe the actual performance of the process%' "hile indices computed from the

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inherent &ariation #"ithin sample si!ma% are referred to as capability indices #since they describe the inherent capability ofthe process%.

Cp an! Cpk

Process Capabilit) etrics• Cp measures ho" "ell the data fits #ithin the spec limits #8S/' /S/%

•

Cpk measures ho" centered the data is bet"een the spec limits.8se Cp Cpk "hen you ha&e a sample' not the population' and are testin! the potential capability of a process to m

customer needs.

Cp and Cpk use Sigma estimator .

Cp, Cpk Formula

#Cp* N 1.55 is desirable%

Pp an! Ppk

Process Performance etrics• Pp measures ho" "ell the data fits #ithin the spec limits #8S/' /S/%

• Ppk measures ho" centered the data is bet"een the spec limits.8se Pp Ppk "hen you ha&e the total populationand are testin! the performance of a system to meet customer

needs.

Pp, Ppk use standard deviation.

Pp, Ppk Formula

#Pp* N 1.55 is desirable%

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Sigma Estimator Formula

d: is a constant based on sub!roup si-e

c4 is a constant based on sub!roup si-e

Lbar ; A&era!e#Li% #A&era!e of the Lan!es in samples%

sbar ; O#i%n

/ne#Si!e! Specifications or 0nilateral Spec 1imits

8se Cp8 #8S/% or Cp/ #/S/% for Cp*.

Capabilit) Ratio # Cr Formula

The Capability Latio #Cr% indicates "hat proportion of tolerance is occupied by the data.

Cr ; 1Cp Cp ; 1.55 Cr ; .0 #data fits 07 of tolerance%

Cpm Formula

Cpm ; #8S//S/%#6R#si!ma:3#KbarTar!et%:%%

#Cpm can be used "hen you ha&e a tar!et &alue.%

 

istakes to 2(oi! 'hen 0sing Cp Cpk an! Pp Ppk Formulas

ou can calculate Cp' Cp* and Pp Pp* manually or you can use the Q ,acros for (cel. ?ased on calls to our tech

support line' most people #ho try to perform manual calculations or build their o#n *)cel formulas end up #

incorrect results. 2or e(ample' they use stde& instead of si!ma estimator for Cp and Cp*. Dr they !et the constant

"ron! based on their sample si-e. f youre pro&idin! these results to your customers shouldnt you use a tool thats

pro&en and  affordableU

3+ acros %ools that Calculate Cp, Cpk an! Pp, Ppk

+istogram  calculates Cp' Cp*' Pp' Pp* and t"enty other metrics usin! your data and spec limits.

Capability Suite  creates si( charts includin! histo!ram' control charts' probability plot' &alues plot and capability p

 Also calculates Cp' Cp* and Pp' Pp*.

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