Cp vs Cpk vs Pp vs Ppk: Differences and Formulas

Differences among the four indices

There are four indices that express process capability: Cp, Cpk, Pp and Ppk. The difference is a combination of the following two points.

• Which standard deviation is used: the within-subgroup standard deviation or the overall standard deviation
• Whether centering is taken into account: not taken into account (Cp, Pp) or taken into account (Cpk, Ppk)

Review: the idea behind Cp and Cpk

Cp and Cpk are indices that show how much margin the variation of a process has relative to the width of the product specification. The larger the value, the less likely out-of-specification product is. In general, a value of 1.33 or more is often used as a target.

• Cp: Looks only at variation. The capability assuming that the mean is at the center of the specification
• Cpk: In addition to variation, also considers that the mean is off the center of the specification

The formulas, worked examples and table of guidelines are explained in detail in What Is Cpk? This article focuses on the other difference, the difference in standard deviation.

Within-subgroup standard deviation and overall standard deviation

Within-subgroup standard deviation (Cp and Cpk)

A set of products made under nearly the same conditions, such as items made consecutively in the same time period, is called a “subgroup.” The within-subgroup standard deviation is the standard deviation calculated from the variation inside the subgroups only. It represents the short-term variation, that is, the potential capability that the process itself has.

The main ways to calculate it are as follows.

• Divide the average R of the subgroup ranges (maximum − minimum) by the coefficient d₂ (R ÷ d₂)
• Combine (pool) the standard deviations of the individual subgroups
• For individual values, divide the average MR of the differences between adjacent values (the moving range) by 1.128 (MR ÷ 1.128)

Overall standard deviation (Pp and Ppk)

The overall standard deviation is the ordinary standard deviation calculated from all the data without splitting it into subgroups. It also includes shifts between subgroups, such as day-to-day changes, differences between lots and materials, and the effects of changeovers. It represents the actual overall performance seen over a long period.

Formulas

Cp and Cpk differ from Pp and Ppk only in the standard deviation used; the form of the formulas is the same. USL is the upper specification limit, LSL is the lower specification limit, and x is the mean.

Worked example

Suppose a process with a specification of 10.0 ± 0.5 (USL = 10.5, LSL = 9.5) and a mean of 10.1 gives the following results.

Using the within-subgroup standard deviation (Cp and Cpk)

• Cp = 1.0 ÷ (6 × 0.08) = 1.0 ÷ 0.48 = 2.08
• Cpk = (10.5 − 10.1) ÷ (3 × 0.08) = 0.4 ÷ 0.24 = 1.67 (the upper side is smaller)

Using the overall standard deviation (Pp and Ppk)

• Pp = 1.0 ÷ (6 × 0.10) = 1.0 ÷ 0.6 = 1.67
• Ppk = (10.5 − 10.1) ÷ (3 × 0.10) = 0.4 ÷ 0.3 = 1.33

Even with the same data, Cpk is 1.67 and Ppk is 1.33. The variation within subgroups is small, but because there are shifts between subgroups, the capability is lower when viewed overall.

What the difference between Cpk and Ppk tells you

Finding both Cpk and Ppk reveals the condition of the process.

• Cpk and Ppk are almost the same: The shift between subgroups is small, and the process is stable
• Cpk is much higher than Ppk: There are shifts between subgroups. Possible causes include day-to-day changes, differences between lots and materials, and changeovers. If the process is stabilized, Ppk can be expected to approach Cpk
• Ppk is much higher than Cpk: This rarely happens. Review how the subgroups are formed (whether items made under different conditions are mixed in one subgroup)

Shifts between subgroups can be checked with the X-bar and R Control Chart (opens in a new tab) or the X-MR Control Chart (Individuals and Moving Range) (opens in a new tab).

Which one to use

Use them according to your purpose.

• You want to know how much out-of-specification product will actually occur: Ppk. It is suited to reporting to customers and evaluating the quality of shipped product
• You want to know the potential capability once the process is stabilized: Cpk. It is useful when considering room for improvement

If your company or customer has a requirement, check which standard deviation it is defined with. For example, in the automotive industry’s PPAP (Production Part Approval Process), Ppk is sometimes used for the initial process study.

When “Cpk” means Ppk: common usage in Japan vs Minitab

In Japan, many companies and textbooks call the values calculated with the overall standard deviation “Cp” and “Cpk.” In Minitab and similar statistical software, these are Pp and Ppk.

Two values called “Cpk” can be calculated differently. When you compare internal reports with software output, or exchange data with a customer, confirm which standard deviation was used and state the calculation method in your report to avoid misunderstandings.

Estimating the within-subgroup standard deviation in Excel

If you use the method based on the range of each subgroup (R ÷ d₂), you can calculate it in Excel too.

1. For each subgroup, find the range (=MAX(…)-MIN(…))
2. Find the average of the ranges, R (=AVERAGE(…))
3. Divide R by the coefficient d₂ that corresponds to the subgroup size n

The overall standard deviation can be found with =STDEV.S(…).

Calculating with the tools on this site

The Process Capability (Cpk) Calculator calculates Cp and Cpk together with Pp and Ppk and displays them side by side, so you can check the difference between them right away. If you specify the subgroup size, it finds the within-subgroup standard deviation; if you do not, it calculates only Pp and Ppk.

The within-subgroup standard deviation is found by pooling the standard deviations of the subgroups when the subgroup size is 2 or more, and from the moving range (MR ÷ 1.128) when it is 1. The value may differ slightly from one found with R ÷ d₂.