Preamble :
This article is devoted to the topic of process capability, with the objective of making people
aware of this subject and its significance to business success.
The author believes that personal awareness is a prerequisite to personal action, and personal action is what we need for success!
 This article can be a source material for you to use in discussing this topic with your organization.
 This article will address issues like what is process capability, how to measure it, and how to calculate the process capability indices (C_{p} and C_{pk}).
 This article will also attempt to explain the differences between process capability and process performance; relationship between C_{pk} and nonconforming (defect) rate; and illustrate the four outcomes of comparing natural process variability with customer specifications.
 Lastly a commentary is provided on precautions we should take while conducting process capability studies.
What is Process Capability ?
 Process capability is the longterm performance level of the process after it
has been brought under statistical control. In other words, process capability is the range
over which the natural variation of the process occurs as determined by the system of common causes.
 Process capability is also the ability of the combination of people, machine, methods, material, and measurements to produce a product that will consistently meet the design requirements or customer expectation.
What is a Process Capability Study ?
Process capability study is a scientific and a systematic procedure that uses control charts to detect and eliminate the unnatural
causes of variation until a state of statistical control is reached. When the study is completed, you will identify the natural variability of the process.
Why Should I know the Capability of My Processes ?
 Process capability measurements allow us to summarize process capability in terms of meaningful percentages and metrics.
 To predict the extent to which the process will be able to hold tolerance or customer requirements. Based on the law of probability, you can compute how often the process will meet the specification or the expectation of your customer.
 You may learn that bringing your process under statistical control requires fundamental changes  even redesigning and implementing a new process that eliminates the sources of variability now at work.
 It helps you choose from among competing processes, the most appropriate one for meeting customers' expectation.
 Knowing the capability of your processes, you can specify better the quality performance requirements for new machines, parts and processes.
Why Should I know the Capability of My Supplier's Processes ?
 To set realistic cost effective part specifications based upon the customer's needs and the costs associated by the supplier at meeting those needs.
 To understand hidden supplier costs. Suppliers may not know or hide their natural capability limits in an effort to keep business. This could mean that unnecessary costs could occur such as sorting to actually meet customer needs.
 To be proactive. For example, a C_{pk} estimation made using injection molding pressure measurements during a molding cycle may help reveal a faulty piston pressure valve ready to malfunction before the actual molded part measurements go out of specifications. Thus saving time and money.
Measures of Process Capability  Process Capability Indices:
C_{p}, C_{pl}, C_{pu} and C_{pk} are the four most common and timed tested measures of process capability.
 Process capability indices measure the degree to which your process produces output that meets the customer's specification.
 Process capability indices can be used effectively to summarize process capability information in a convenient unitless system.
 C_{p} and C_{pk} are quantitative expressions that personify the variability of your process (its natural limits) relative to its specification limits (customer requirements).
Following are the graphical details and equations quantifying process capability:

Where:
USL = Upper Specification Limit
LSL = Lower Specification Limit
XBar = Mean of the Process
s = Standard Deviation of the Process

Index 
Estimated Equation 
Usage 
C_{p} 
( USL  LSL ) / 6s 
Process Capability for two  sided specification limit, irrespective of process center. 
C_{pu} 
( USL  XBar ) / 3s 
Process Capability relative to upper specification limit. 
C_{pl} 
( XBar  LSL ) / 3s 
Process Capability relative to lower specification limit. 
C_{pk} 
Min. of ( C_{pu} , C_{pl} ) or ( Distance between mean of the process and the closest spec. limit / 0.5 of the process variability ) 
Process Capability for two  sided specification limit accounting for process centering. 
Notes :
 If XBar is at target, then C_{p} = C_{pk}.
 C_{pk} will always be equal to or less than C_{p}.
The C_{pk}, P_{pk} Quandary :
In 1991, ASQ / AIAG task force published the "Statistical Process Control" reference manual, which presented the calculations for capability indices ( C_{p}, C_{pk} )
as well as process performance indices ( P_{p}, P_{pk} ).
The difference between the two indices is the way the process standard deviation ( s ) is calculated.
C_{pk} uses s which is estimated using ( RBar / d_{2} ) or ( SBar / C_{2} ) .
P_{pk} uses the calculated standard deviation from individual data where s is calculated by the formula :
So the next question is which metric is best to report C_{pk} or P_{pk} ? In other words, which standard deviation to use  estimated or calculated ?
Although both indices show similar information, they have slightly different uses.
 P_{pk} attempts to answer the question "does my current production sample meet specification ?" Process performance indices should only be used when statistical control cannot be evaluated.
 On the other hand, C_{pk} attempts to answer the question "does my process in the long run meet specification?" Process capability evaluation can only be done after the process is brought into statistical control.
The reason is simple: C_{pk} is a prediction, and one can only predict something that is stable.
The readers should note that P_{pk} and C_{pk} indices would likely be similar when the process is in a state of statistical control.
Notes :
 As a thumb rule a minimum of 50 randomly selected samples must be chosen for process performance studies and a minimum of 20 subgroups ( of sample size, preferably of at least 4 or 5 ) must be chosen for process capability studies.
 C_{pk} for all critical product measurements considered important by the customer should be calculated at the beginning of initial production to determine the general ability of the process to meet customer specifications. Then from time to time, over the life of the product, C_{pk}s must be generated.
A control chart must always be maintained to check statistical stability of the process before capability is computed.
Process Capability and Defect Rate :
Using process capability indices it is easy to forget how much of product is falling beyond specification. The conversion curve presented here can be a useful tool for interpreting C_{pk} with its corresponding defect levels.
The defect levels or parts per million nonconforming were computed for different C_{pk} values using the Z scores and the percentage area under the standard normal curve using normal deviate tables.
The table below presents the nonconforming parts per million ( ppm ) for a process corresponding to C_{pk} values if the process mean were at target.
C_{pk} Value 
Sigma Value 
Area under Normal Curve 
Non Conforming ppm 
0.1  0.3 
0.235822715  764177.2851 
0.2  0.6  0.451493870  548506.1299 
0.3  0.9 
0.631879817  368120.1835 
0.4  1.2 
0.769860537  230139.4634 
0.5  1.5 
0.866385542  133614.4576 
0.6  1.8 
0.928139469  71860.531 
0.7  2.1 
0.964271285  35728.7148 
0.8  2.4 
0.983604942  16395.0577 
0.9  2.7 
0.993065954  6934.0461 
1.0  3.0 
0.997300066  2699.9344 
1.1  3.3 
0.999033035  966.9651 
1.2  3.6 
0.999681709  318.2914 
1.3  3.9 
0.999903769  96.231 
1.333  3.999 
0.999936360  63.6403 
1.4  4.2 
0.999973292  26.7082 
1.5  4.5 
0.999993198  6.8016 
1.6  4.8 
0.999998411  1.5887 
1.666  4.998 
0.999999420  0.5802 
1.7  5.1 
0.999999660  0.3402 
1.8  5.4 
0.999999933  0.0668 
1.9  5.7 
0.999999988  0.012 
2.0  6.0 
0.999999998  0.002 
The C_{pk} conversion curve for process with mean at target is shown next.
Explanation :
A process with C_{pk} of 2.0 ( +/ 6 sigma capability), i.e., the process mean is 6 sigma away from the nearest specification can be expected to have no more than 0.002 nonconforming parts per million.
This process is so good that even if the process mean shifts by as much as +/ 1.5 sigma the process will produce no more than 3.4 nonconforming parts per million.
The next section provides the reader with some practical clarifications on: Process Capability ( Voice of the process ) and Specification ( Expectations of the customer ).
Natural Variability versus Specifications for Process Capability :
As seen from the earlier discussions, there are three components of process capability:
 Design specification or customer expectation ( Upper Specification Limit, Lower Specification Limit )
 The centering of the natural process variation ( XBar )
 Spread of the process variation ( s )
A minimum of four possible outcomes can arise when the natural process variability is compared with the design specifications or customer expectations:
Case 1: C_{pk} > 1.33 ( A Highly Capable Process )
This process should produce less than 64 nonconforming ppm
A Highly Capable Process : Voice of the Process < Specification ( or Customer Expectations ).
This process will produce conforming products as long as it remains in statistical control.
The process owner can claim that the customer should experience least difficulty and greater reliability with this product. This should translate into higher profits.
Note : C_{pk} values of 1.33 or greater are considered to be industry benchmarks. This means that the process is contained within four standard deviations of the process specifications.
Case 2: C_{pk} = 1 to 1.33 ( A Barely Capable Process )
This process will produce greater than 64 ppm but less than 2700 nonconforming ppm.
A Barely Capable Process : Voice of the Process = Customer Expectations.
This process has a spread just about equal to specification width. It should be noted that if the process mean moves to the left or the right,
a significant portion of product will start falling outside one of the specification limits. This process must be closely monitored.
Note : This process is contained within three to four standard deviations of the process specifications.
Case 3: C_{pk} < 1 ( The Process is not Capable )
This process will produce more than 2700 nonconforming ppm.
A NonCapable Process : Voice of the Process > Customer Expectations.
It is impossible for the current process to meet specifications even when it is in statistical control.
If the specifications are realistic, an effort must be immediately made to improve the process (i.e. reduce variation) to the point where it is capable of producing consistently within specifications.
Case 4: C_{pk} < 1 ( The Process is not Capable )
This process will also produce more than 2700 nonconforming ppm.
OffCenter & NonCapable Process.
The variability ( s ) and specification width is assumed to be the same as in case 2, but the process average is offcenter. In such cases, adjustment is required to move the process mean back to target.
If no action is taken, a substantial portion of the output will fall outside the specification limit even though the process might be in statistical control.
Assumptions, Conditions and Precautions :
Capability indices described here strive to represent with a single number the capability of a process. Much has been written in the literature about the pitfalls of these estimates.
Following are some of the precautions the readers should exercise while calculating and interpreting process capability:
 The indices for process capability discussed are based on the assumption that the underlying process distribution is approximately bell shaded or normal. Yet in some situations the underlying process distribution may not be normal.
For example, flatness, pull strength, waiting time, etc., might natually follow a skewed distribution. For these cases, calculating C_{pk} the usual way might be misleading. Many researchers have contributed to this problem.
Readers are requested to refer to John Clements article titled
"Process Capability Calculations for NonNormal Distributions" for details.
 The process / parameter in question must be in statistical control. It is this author's experience that there is tendency to want to know the capability of the process before statistical control is established.
The presence of special causes of variation make the prediction of process capability difficult and the meaning of C_{pk} unclear.
 The data chosen for process capability study should attempt to encompass all natural variations.
For example, one supplier might report a very good process capability value using only ten samples produced on one day,
while another supplier of the same commodity might report a somewhat lesser process capability number using data from longer period of time that more closely represent the process. If one were to compare
these process index numbers when choosing a supplier, the best supplier might not be chosen.
 The number of samples used has a significant influence on the accuracy of the C_{pk} estimate.
For example, for a random sample of size n = 100 drawn from a know normal population of C_{pk} = 1,
the C_{pk} estimate can vary from 0.85 to 1.15 ( with 95 % confidence ). Therefore smaller samples will result in even larger variations of the C_{pk} statistics.
In other words, the practitioner must take into consideration the sampling variation's influence on the computed C_{pk} number. Please refer to Bissell and Chou, Owen, and Borrego for more on this subject.
Concluding Thoughts :
In the real world, very few processes completely satisfy all the conditions and assumptions required for estimating C_{pk}.
Also, statistical debates in research communities are still raging on the strengths and weaknesses of various capability and performance indices.
Many new complicated capability indices have also been invented and cited in literature. However, the key to effectual use of process capability measures continues to be the level of user understanding of what these measures really represent.
Finally, in order to achieve continuous improvement, one must always attempt to refine the "Voice of the Process" to match and then to surpass the "Expectations of the Customer".
References :
 Victor Kane, "Process Capability Indices", Journal of Quality Technology, Jan 1986.
 ASQ / AIAG, "Statistical Process Control", Reference Manual, 1995.
 John Clements, "Process Capability Calculations for NonNormal Distributions", Quality Progress, Sept 1989.
 Forrest Breyfogle, "Measurement of Process Capability", Smarter Solutions, 1996.
 Bissell, "How Reliable is Your Capability Index", Royal Statistical Society, 1990.
 Chou, Owen, and Borrego, "Lower Confidence Limits of Process Capability Indices",
Journal of Quality Technology, Vol 22, No. 3, July 1990.
Author: Dr. Mehernosh Kapadia
Published: November 2000

Dr. Mehernosh Kapadia is General Manager  Quality Engineering, Supply Chain Management Business Group at Tata AutoComp Systems Limited.
He has a Master's Degree in Mechanical Engineering from Penn State University, USA and MBA from NTU, USA.
He is a SixSigma Master Black Belt and an ASQ certified CQE & CRE.
He obtained his Doctorate from The University of Bombay
He can be contacted through us at info@symphonytech.com

