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Cp Cpk Pp Ppk and % out of tolerance

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To check if the production process is capable to produce within specification limits. The Cp Cpk Pp and Ppk are used together with Control charts in Statistical process control (SPC)

Cp Cpk

The Cp and Cpk is calculated when the subgroup is set 1 or higher . When this value is 0 the Pp and Ppk are calculated.

The Cp and Cpk is used when a process is under statistical control. For not production environment (testing, new process or setting) use the Pp and Ppk!

Cp (Process Capability if centered)

The Cp indicates if a process is capable to produce within the specification limits, if the process is centered between the specification limits. The Cp does not include the drift between the subgroups! If you want to included the drift use the Pp. To get this quality (Cpk) the process needs to be tuned so that mean is moved to center of the tolerance borders. Therefore the Cp is always lower of equal compared to the Cpk value. If the cell with the Cp value is
Violet
the process is probably not in control the difference between Cp and Pp is bigger as 15%.

The Cp value of a data-set is only calculated if the min and max tolerance are available.

Formula

formula
With a higher Cp value the variation is smaller and it is possible to produce with a higher quality.

Cpk (Process Capability Index)

Calculates the Process capability value of a data-set. The Cpk does not include the drift between the subgroups! If you want to included the drift use the Ppk. The higher the Sigma level the higher the quality (see table). If the cell with the Cpk value is
Violet
the process is probably not in control the difference between Cpk and Ppk is bigger as 15%.

Formula

formula
With a higher Cpk value the data set is better within tolerance.

Sigma level table

Two sided table
Cpk Ppk Sigma level % out of tolerance PPM out of tolerance
0.33 1.0 31.73 317310.508
0.50 1.5 13.36 133614.403
0.67 2.0 4.55 45500.264
0.83 2.5 1.24 12419.331
1.00 3.0 0.27 2699.796
1.17 3.5 0.05 465.258
1.33 4.0 0.01 63.342
1.50 4.5 0.001 6.795
1.67 5.0 0.0001 0.573
1.83 5.5 0.000004 0.038
2.00 6.0 0.0000002 0.002


Subgroups

0 is no subgroups the overall STDEV will be used and the Pp and Ppk will be calculated.
1 is no subgroups the STDEV within will be estimated and the Cp and Cpk will be calculated.
>1 there are subgroups and the Cp and Cpk are calculated with the estimated STDEV within.

What is a subgroup?

If in a production process after every hour 5 measurements are done, the subgroup size is 5. When calculating the STDEV within data is ordered in groups of 5 to filter out the drift between the different groups.

So the the following data is filtered But it includes the short therm variation of the process.

How to use subgroups?

When subgroup size of 5 is selected then row 1 to 5 is for subgroup 1 and row 6 to 10 is for subgroup 2 etc.

Data file

Legend

formula = Mean
USL = Upper specification limit
LSL = Lower specification limit
formula = STDEV within

Minimum sample size Cp and Cpk

The minimum sample size to estimate the Cp and Cpk is 30 samples.

What if the process is not approximately normally distributed

The Cp and Cpk assume that the data is according a normal distribution, but due use of subgroups less vulnerable for non normal overall data. The normally check is calculated on the overall data and does not use the subgroups!

What if the process is not in control

  1. Do a Gauge R&R and try to improve the measurement.
  2. Label the data in subgroups to identify the factors that can cause that the process is not in control.

Pp Ppk

The Pp and Ppk is calculated when the subgroup is set 0 . When the value is higher the Cp and Cpk are calculated. The Pp and Ppk are generally lower as the Cp and Cpk values.

Pp (Process Performance if centered)

The Pp indicates if a process is capable to produce within the specification limits, if the process is centered between the specification limits. To get this quality (Ppk) the process needs to be tuned so that mean is moved to center of the tolerance borders. Therefore the Pp is always lower of equal compared to the Ppk value. This calculation assumes that the data is normally distributed this can be checked with the Anderson Darling normality test. If the cell with the Pp value is
Red
the data set is not according the normal distribution.
The Pp value of a data-set is only calculated if the min and max tolerance are available.

Formula

formula
With a higher Pp value the variation is smaller and it is possible to produce with a higher quality.

Ppk (Process Performance Index)

Calculates the Process capability value of a data-set. The higher the Sigma level the higher the quality (see table). This calculation assumes that the data is normally distributed this can be checked with the Anderson Darling normality test. If the cell with the Ppk value is
Red
the data set is not according the normal distribution.

Formula

formula
With a higher Ppk value the data set is better within tolerance.

Minimum sample size Pp and Ppk

The minimum sample size to estimate the Pp and Ppk is 30 samples.

% out of tolerance

Calculates how much % of the data set is statistical out of tolerance. Below 0.1% the value is displayed in PPM (Parts Per Million). This calculation assumes that the data is normally distributed this can be checked with the Anderson Darling normality test. If the cell with the "% out of tolerance" value is
Red
the data set is not according the normal distribution.

Formula

formula calculate the formula value out of the t distribution.
formula calculate the formula value out of the t distribution.

% out of tolerance formula

Legend

formula = Mean
s = STDEV
USL = Upper specification limit
LSL = Lower specification limit

What if the process is not approximately normally distributed

The Pp, Ppk and % out of tolerance assume that the data is according a normal distribution. This can be checked with the Anderson Darling normality test. If the data set is not normally distributed see "What to do with not normally distributed data". A solution is to transform the data and tolerance limits with a Box-Cox transformation. If the result of this transformation is a normally distributed data-set the Pp, Ppk and % out of tolerance can be used.

Example

From data-set in column A the CP is 1.17, Cpk 1.03 and statistical is 0.11% out of tolerance, this is visible in the rows: Cp, Cpk and % out of tol of the result array. The values are calculated with the tolerance borders in Max Tol. and Min Tol.

Data file

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