Cp Cpk Formulas versus Pp Ppk Formulas

Formulas for Cp Cpk and Pp Ppk are Similar

People often get confused about the difference between Pp/Ppk and Cp/Cpk.

  • Pp/Ppk always use standard deviation σ as the divisor, not sigma estimator (σ with a hat).
  • Individuals data always uses Rbar/d2 to estimate standard deviation.
  • Samples of 2 or more: Cp/Cpk used to use Rbar/d2 or Sbar/c4 to estimate sigma. Cp/Cpk now use pooled standard deviation although the difference is small, often less than 0.005.

Cp, Cpk Formula & Calculations

Cp, Cpk formulas

(Cpk > 1.33 is desirable)

Cp and Cpk use Sigma Estimator.

Pp, Ppk Formula & Calculations

Pp, Ppk formulas

(Ppk > 1.33 is desirable)

Pp, Ppk use standard deviation.

One-Sided or Unilateral Specification Limits

Use CpU (USL) or CpL (LSL) for Cpk.

Thinking About Trying to Do This Yourself? Don't. Here's Why:

Time: You will probably spend hours trying to figure it out. Why not download a free trial of the QI Macros and start doing it correctly immediately?

Multiple Complicated Formulas to Choose From: You can try to calculate Cp, Cpk and Pp Ppk manually, but you'll probably make mistakes and aggravate customers. And you will find your homegrown template hard to maintain.

  • Based on calls to our tech support line, MOST people who try to perform manual calculations or build their own Excel formulas end up with incorrect results. Then they question our results!
  • For example, they use standard deviation instead of sigma estimator for Cp and Cpk. Or they get the constant wrong based on their sample size. Or they don't understand there are several methods of calculating sigma estimator.

If you're providing these results to your customers shouldn't you use a tool that's proven and affordable?

Sigma Estimator - There are Three Different Ways to Estimate Standard Deviation

n represents the subgroup size in these Sigma Estimator formulas

Pooled Standard Deviation

Use when n > 2
Cp and Cpk Pooled Standard Deviation Formulas - QI Macros

Average of the Subgroup Standard Deviations

Use when n > 4
c4 is a constant based on subgroup size
sbar = Σ(σi)/n

sbar formula to estimate standard deviation

Average of the Subgroup Ranges

Use when n = 1 to 4
d2 is a constant based on subgroup size
Rbar = Average(Ri) (Average of the Ranges in samples)

sigma estimator using Rbar / d2

Sigma Estimator Calculation Used in QI Macros

QI Macros versions dated after May 2015 use these defaults:

  • Pooled Standard Deviation when n >= 2 and
  • Rbar/d2 when n = 1.

Users can change the sigma estimator calculation after they have run a histogram by changing the estimator fields on the far right of the histogram data sheet.

sigma estimator calculation in QI Macros

In addition between changing from Pooled Standard deviation to Sbar or Rbar, users can also choose Between/Within Deviation. This option was added to QI Macros in December 2015.

Note: Minitab started using Pooled Standard Deviation to calculate Cp/Cpk, and control limits on XbarR and XbarS charts in versions 15 and 16. Minitab 17 went back to Rbar/d2 and Sbar/c4 for XbarR/S control limits, but retained pooled stdev for Cp/Cpk calculations when using multiple samples.

Pp and Ppk Calculations in Weibull Histogram

Unlike the Pp and Ppk calculations which rely on a Normal distribution, the Weibull histogram uses the WEIBULL (or WEIBULL.DIST) function to calculate the Z-scores for the USL/LSL and from these it cacluates Pp and Ppk. There is no Cp or Cpk calculation possible in the Weibull Histogram.

If there is only a USL or LSL, the Weibull histogram Ppk is either PpU or PpL.

QI Macros Tools that Calculate Cp, Cpk and Pp, Ppk

  • Histogram - calculates Cp, Cpk, Pp, Ppk and 20 other metrics using your data and spec limits.
  • Weibull Histogram - calculates Pp, Ppk and other metrics using your data and spec limits.
  • Capability Suite - creates six charts including histogram, control charts, probability plot, values plot and capability plot. Also calculates Cp, Cpk and Pp, Ppk.
  • Cp Cpk Template - calculates Cp, Cpk and Pp, Ppk on multiple sets of data.

Learn More ...

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