What Is Cpk? Formula, Worked Example and Target Values
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What is Cpk?
Cpk (process capability index) is a number that shows how much margin a process has to make items within the specification, relative to the width of the specification. The larger the value, the less likely out-of-specification product is.
How good a process is depends on the following two things.
- Variation: how widely the measured values are scattered
- Centering: how far the mean of the measured values is from the center of the specification
Cpk combines these two into a single number.
Difference between Cp and Cpk
A similar index is Cp. The difference is whether the offset of the center is taken into account.
- Cp: Looks only at variation. The capability assuming that the mean is at the center of the specification
- Cpk: Considers the offset of the mean in addition to variation. The actual capability
If the mean is exactly at the center of the specification, Cp and Cpk have the same value. The closer the mean is to either the upper or the lower limit, the smaller Cpk becomes than Cp. When Cp is high but Cpk is low, you can tell that the problem is not the variation but an “off-center mean.”
Process conditions seen in histograms
Laying the upper and lower specification limits over a histogram of the measured data shows the condition of the process at a glance. Let us compare four common conditions for a process with a specification of 10.0 ± 0.5.
Within specification Out of specification (the light bands are outside the specification)
- A: Good condition: The peak is at the center of the specification, and the width is narrow enough relative to the specification. Both Cp and Cpk are 1.67
- B: Mean is off-center: The variation is the same as in A, but the peak is shifted toward the upper limit. Even though Cp remains 1.67, Cpk falls to 0.83, and some values exceed the upper limit. The remedy is to bring the mean back to the center of the specification (for example, by adjusting the equipment)
- C: Large variation: The peak is at the center, but it is wide, and values exceed the specification on both sides. Both Cp and Cpk are 0.83. The remedy is to reduce the causes of the variation
- D: Both off-center and large variation: The mean is off-center and the variation is also large. Cp is 0.83 and Cpk is 0.50. Both the offset and the variation need to be addressed
Comparing Cp and Cpk shows whether the problem lies in the offset or in the variation. If Cp is high and only Cpk is low, the main cause is an off-center mean (B); if Cp is low too, the main cause is variation (C and D).
Formulas
The symbols are as follows.
| Symbol | Meaning |
|---|---|
| USL | Upper specification limit |
| LSL | Lower specification limit |
| Mean of the measured values | |
| σ | Standard deviation |
Cp = (USL − LSL) ÷ 6σ
CPU = (USL − ) ÷ 3σ (margin on the upper side)
CPL = ( − LSL) ÷ 3σ (margin on the lower side)
Cpk = the smaller of CPU and CPL
The idea is that the capability of the process is determined by whichever of the upper and lower limits is closer to the mean (the one with less margin).
Calculating with the K factor
The same Cpk can also be calculated using K, a value that expresses how far the mean is from the center of the specification (the degree of offset).
K = |specification center − | ÷ (specification width ÷ 2)
(Specification center = (USL + LSL) ÷ 2, specification width = USL − LSL)
Cpk = (1 − K) × Cp
K is 0 when the mean is at the center of the specification, and 1 when the mean is at the upper or lower limit.
Worked example
Let us calculate with the following conditions.
| Item | Value |
|---|---|
| Specification | 10.0 ± 0.5 (USL = 10.5, LSL = 9.5) |
| Mean | 10.1 |
| Standard deviation σ | 0.1 |
- Cp = (10.5 − 9.5) ÷ (6 × 0.1) = 1.0 ÷ 0.6 = 1.67
- CPU = (10.5 − 10.1) ÷ (3 × 0.1) = 0.4 ÷ 0.3 = 1.33
- CPL = (10.1 − 9.5) ÷ (3 × 0.1) = 0.6 ÷ 0.3 = 2.00
- Cpk = the smaller of 1.33 and 2.00 = 1.33
Because the mean is shifted 0.1 toward the upper limit, Cpk is smaller than Cp.
You can also check this using K.
- K = |10.0 − 10.1| ÷ (1.0 ÷ 2) = 0.1 ÷ 0.5 = 0.2
- Cpk = (1 − 0.2) × 1.67 = 1.33 (the same value)
When there is only one specification limit
Some specifications have only an upper or only a lower limit, such as “X or less” or “X or more.” In this case, Cp cannot be calculated, and the value for the side that exists is used as Cpk as is.
- Upper limit only: Cpk = CPU
- Lower limit only: Cpk = CPL
Note that Cpk becomes negative if the mean is outside the specification.
Cpk guidelines
The guidelines commonly used for judgment are as follows.
| Cpk | Guideline | Share outside the nearer specification limit (normal distribution) |
|---|---|---|
| 1.67 or more | More than sufficient. Simplifying controls or reviewing costs can also be considered | About 0.00003% (0.3 ppm) |
| 1.33 to less than 1.67 | Sufficient | About 0.003% (32 ppm) |
| 1.00 to less than 1.33 | Acceptable. Be careful as it approaches 1.00 | About 0.13% (1,350 ppm) |
| 0.67 to less than 1.00 | Insufficient. Improvement is needed | More than 0.13% |
| Less than 0.67 | Very insufficient | — |
Note: The percentage is the share that falls outside the specification limit nearer to the mean.
Many companies aim for a Cpk of 1.33 or more, but the required value depends on the customer, the industry, and the importance of the product. If your customer specifies a requirement, follow it.
Points to note when calculating
Cpk is a convenient number, but unless the following conditions are met, it may deviate from the actual capability of the process.
- Number of data points: With few data points the value is unstable. A commonly cited guideline is at least 30, preferably 50 to 100 or more
- Shape of the distribution: The calculation assumes a normal distribution. It is reassuring to check on a histogram that the shape has a single peak and is close to symmetrical
- Process stability: If the mean or the variation changed partway through, Cpk loses its meaning. The basic approach is to first check on a control chart that the process is stable
The Histogram Tool (opens in a new tab) and the X-bar and R Control Chart (opens in a new tab) are also free to use on this site.
How the standard deviation for “Cpk” is calculated
There are two ways of thinking about the standard deviation used to calculate Cpk.
- Use the within-subgroup standard deviation (the variation inside a subgroup over a short period)
- Use the overall standard deviation
In Japan, many companies and textbooks call the value calculated with the overall standard deviation “Cpk.” Statistical software such as Minitab, on the other hand, calls the value calculated with the within-subgroup standard deviation Cpk and the value calculated with the overall standard deviation Ppk. The formulas for Pp and Ppk are the same as those for Cp and Cpk; the only difference is that σ is the overall standard deviation. The tools on this site also display both values, following this naming. When you compare against your company’s standard, check which standard deviation it uses.
The differences among Cp, Cpk, Pp and Ppk, and which to use, are explained in detail in Cp vs Cpk vs Pp vs Ppk.
Calculating in Excel
This is an example where the data are in A2 to A51, the upper specification limit is in E1, and the lower limit is in E2.
| What to calculate | Example formula |
|---|---|
| Mean | =AVERAGE(A2:A51) (enter in E3) |
| Standard deviation | =STDEV.S(A2:A51) (enter in E4) |
| Cp | =(E1-E2)/(6*E4) |
| Cpk | =MIN((E1-E3)/(3*E4),(E3-E2)/(3*E4)) |
This formula uses the overall standard deviation, so it corresponds to the “Cpk” commonly used in Japan, as explained above, and to Ppk in statistical software.
If you want to paste in your data and get Cp, Cpk, Pp, Ppk and a histogram all at once, the Process Capability (Cpk) Calculator on this site is convenient. No registration is required, and it is free to use.
Open the Process Capability (Cpk) Calculator (opens in a new tab)