Process capability index

The process capability indices Cp and CpK are key figures for the statistical evaluation of a process in production technology . They indicate how reliably the goals specified in the specification can be achieved.

definition

The following formulas only apply to normally distributed characteristics. In DIN ISO 22514-2 (formerly DIN ISO 21747) there are calculation methods that can be used for all distribution models.

The C _pK value is defined as follows from the mean value , the associated standard deviation and the upper ( ) or lower ( ) specification limit: ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle OSG}$ ${\displaystyle USG}$

${\displaystyle C_{pK}={\frac {\min(\mu -USG,OSG-\mu )}{3\sigma }}.}$

The higher this value, the more certain the entire production is within the specification.

The C _p value is defined as:

${\displaystyle C_{p}={\frac {OSG-USG}{6\sigma }}.}$

The C _p value can only be calculated if both an upper and a lower specification limit are defined.

While the C _p value only indicates the ratio of the specified tolerance to the process variance, the C _pK value also includes the position of the mean value in relation to the specified tolerance center. In the best case (process mean value is exactly in the middle of the tolerance range), C _pK = C _p ; otherwise C _pK <C _p .

The individual letters of the abbreviation stand for:

• C: Capability
• p: process
• K: Katayori (Japanese), which means something like "bias" or "displacement".  

Target values

In the past, a C _pK value of at least 1.00 (the distance between the closest tolerance limit and the process mean value is at least 3 standard deviations) was considered sufficient, later the requirement was raised to 1.33 (4 standard deviations). In the meantime, a C _p value of 2.00 (the width of the tolerance range corresponds to a spread of ± 6 standard deviations, hence Six Sigma ) is combined with a C _pK value of 1.67 (the distance between the closest tolerance limit and the process mean is at least 5 standard deviations) as a desirable goal. It should be noted that an overall system made up of many components naturally has a significantly higher error rate than the individual components. For a sufficiently high C _pK value of the system, the components must have an even higher value.

Comparison _table C _pK - PPM

Assuming a normally distributed process variable, the number of expected errors per 1 million ( parts per million ) can be calculated from the process capability index C _pK using the following formula :

${\displaystyle PPM=(1-F(3\cdot c_{pK})+F(-3\cdot c_{pK}))\cdot 1\,000\,000=2\cdot F(-3\cdot c_{pK})\cdot 1\,000\,000}$

It is the distribution function of the standard normal distribution . The following table gives some example values ​​for the two-sided probability: ${\displaystyle F}$

C pK	PPM	Sigma
0.50	133614
0.67	045500
0.79	020000
0.90	006933
1.00	002699	3σ
1.30	000096
1.33	000066	4σ
1.42	000020th
1.50	000003.4
1.60	000002
1.67	000000.6	5σ
2.00	000000.002	6σ

If the assumption of a normal distribution is not fulfilled, then the relationship between the index and the error rate in PPM is different. Often data are e.g. B. log-normal or Student-T distributed. In both of these cases, the PPM value for an index of 1.5, for example, is much smaller than for a normal distribution; the opposite would be the case with a rectangular distribution (uniform distribution). For this reason, a normality test should be carried out. However, this can be imprecise if the amount of data is too small.

criticism

The Cp and Cpk values ​​have a meaning if there is a normal distribution. The simplest means of increasing the process capability of a given process is to loosen the specification limits: the greater the difference between the USL and ESL, the more standard deviations can be accommodated in it. The elimination of specification limits means that an infinite process capability is achieved.

In order for the process capability to remain a reasonable level, the specification limits must in no case be influenced by the process owner.

The higher the required Cpk value, the less the feature tolerances given in the drawing, for example, can be used.

See also

• Process capability study
• Machine capability

literature

• Stephan Lunau (eds.), Olin Roenpage, Christian Staudter, Renata Meran, Alexander John, Carmen Beernaert: Six Sigma + Lean Toolset: Successful implementation of improvement projects 2nd, revised edition. Springer, ISBN 3-540-46054-3
• Standard DIN ISO 3534-2: 2013: Statistics - Terms and symbols - Part 2: Applied statistics
• Standard DIN ISO 22514-2: 2015: Statistical methods in process management - Capability and performance - Part 2: Process performance and process capability parameters of time-dependent process models

Web links

• Free Excel template for calculating process capability

Individual evidence

1. ↑ Process Capability (Cp, Cpk) and Process Performance (Pp, Ppk) - What is the Difference? - iSixSigma. February 26, 2010. Retrieved October 14, 2019 (American English). 
2. ↑ Dietrich, Schulze: Statistical procedures for machine and process qualification . 5th edition, p. 186 .
3. ↑ Thomas Pyzdek: Motorola's Six Sigma program . (English).
4. ↑ Keki R. Bhote: The power of ultimate Six Sigma . AMACOM Div American Mgmt Assn, 2003, p. 19.