A process capability index tells you whether a stable process can keep producing parts within specification. It compares the voice of the customer – the lower and upper specification limits (LSL and USL) – with the voice of the process – its natural spread of about six standard deviations.
Cp looks only at spread: it is the tolerance width divided by 6σ, so Cp = 1 means the process just fits if it is perfectly centred. Cpk also accounts for where the mean sits: it measures the distance from the mean to the nearer limit in units of 3σ, so an off-centre process gets a lower Cpk than its Cp. Most industries ask for Cpk ≥ 1.33 (a four-sigma margin), critical characteristics often 1.67, and a six-sigma process has Cpk = 2.
When there is a target value T, often the middle of the tolerance, the Taguchi index Cpm also penalises being off target, and Cpkm combines both ideas. The calculator also translates Cpk into the sigma level and, assuming normal data, estimates the parts per million outside each limit.
How to use the Process Capability Index Calculator
- Upper specification limit (USL)
- Lower specification limit (LSL)
- Process mean (μ)
- Process standard deviation (σ)
- Target value
- Target value (T)
Results update as you type, and everything is calculated in your browser.
How it works
Enter the specification limits and the process mean and standard deviation (from a stable control chart or a capability study of at least 30–50 parts). Optionally add a target value. The calculator applies the formulas above and uses the normal distribution to estimate the fraction outside the limits.
Worked example
A shaft must measure 9.5–10.5 mm. The process mean is 10.1 mm and σ = 0.12 mm. Cp = 1.0 ÷ 0.72 = 1.389, Cpu = 0.4 ÷ 0.36 = 1.111 and Cpl = 0.6 ÷ 0.36 = 1.667, so Cpk = 1.111 – only marginal, because the mean sits 0.1 mm above the 10 mm target. About 429 ppm are expected above the USL and 0.29 ppm below the LSL. With T = 10, Cpm = 1 ÷ (6 × √(0.0144 + 0.01)) = 1.067 and Cpkm = 1.111 ÷ √(1 + 0.694) = 0.854. Re-centring the process at 10.0 would raise Cpk to 1.389.
Cpk, sigma level and defects
| Cpk | Sigma level | Out of spec (one side, normal) |
|---|---|---|
| 0.67 | 2σ | 22,750 ppm |
| 1.00 | 3σ | 1,350 ppm |
| 1.33 | 4σ | 32 ppm |
| 1.67 | 5σ | 0.29 ppm |
| 2.00 | 6σ | 0.001 ppm |
Inputs explained
| Field | What to enter |
|---|---|
| Upper specification limit (USL) | A number |
| Lower specification limit (LSL) | A number |
| Process mean (μ) | A number |
| Process standard deviation (σ) | A number |
| Target value | Choose one: No target – Cp and Cpk only, Given a target value T |
| Target value (T) | A number |
What each result means
- Potential capability Cp = ([Upper specification limit (USL)] – [Lower specification limit (LSL)])÷(6×[Process standard deviation (σ)])
- Upper capability Cpu = ([Upper specification limit (USL)] – [Process mean (μ)])÷(3×[Process standard deviation (σ)])
- Lower capability Cpl = ([Process mean (μ)] – [Lower specification limit (LSL)])÷(3×[Process standard deviation (σ)])
- Process capability index Cpk = min([Upper capability Cpu], [Lower capability Cpl])
- Taguchi index Cpm = ([Upper specification limit (USL)] – [Lower specification limit (LSL)])÷(6×sqrt([Process standard deviation (σ)]^2 + ([Process mean (μ)] – [Target value (T)])^2))
- Cpkm = [Process capability index Cpk]÷sqrt(1 + (([Process mean (μ)] – [Target value (T)])÷[Process standard deviation (σ)])^2)
- Sigma level (3 × Cpk) = 3×[Process capability index Cpk]
- Off-centre index k = abs(([Upper specification limit (USL)] + [Lower specification limit (LSL)])÷2 – [Process mean (μ)])÷(([Upper specification limit (USL)] – [Lower specification limit (LSL)])÷2)
- Expected above USL = (1 – ncdf(([Upper specification limit (USL)] – [Process mean (μ)])÷[Process standard deviation (σ)]))×1000000
- Expected below LSL = ncdf(([Lower specification limit (LSL)] – [Process mean (μ)])÷[Process standard deviation (σ)])×1000000
- Expected out of specification = [Expected above USL] + [Expected below LSL]
- Expected yield within specification = 100 – [Expected out of specification]÷10000
Formula
Cp = (USL − LSL) ÷ 6σ · Cpu = (USL − μ) ÷ 3σ · Cpl = (μ − LSL) ÷ 3σ · Cpk = min(Cpu, Cpl) · Cpm = (USL − LSL) ÷ 6√(σ² + (μ − T)²) · Cpkm = Cpk ÷ √(1 + ((μ − T) ÷ σ)²)
Worked example
Inputs — Upper specification limit (USL): 10.5; Lower specification limit (LSL): 9.5; Process mean (μ): 10.1; Process standard deviation (σ): 0.12; Target value: Given a target value T; Target value (T): 10.
Results — Potential capability Cp = 1.3889; Upper capability Cpu = 1.1111; Lower capability Cpl = 1.6667; Process capability index Cpk = 1.1111; Rating = Marginal (1.00 – 1.33); Taguchi index Cpm = 1.067; Cpkm = 0.8536; Sigma level (3 × Cpk) = 3.333; Off-centre index k = 0.2; Expected above USL = 429.06 ppm; Expected below LSL = 0.29 ppm; Expected out of specification = 429.35 ppm; Expected yield within specification = 99.95707 %.
Key terms
- Specification limits
- The customer's boundaries (LSL and USL) for an acceptable product.
- Cp
- The tolerance width divided by six process standard deviations.
- Cpk
- The distance from the mean to the nearer limit in units of three standard deviations.
- ppm
- Parts per million – expected defects out of one million units.
Frequently asked questions
What is a good Cpk value?
1.33 is the usual minimum, 1.67 is expected for safety-critical or new processes, and 2.0 corresponds to six-sigma capability. Below 1.0 the process makes out-of-spec parts even when everything is normal.
What is the difference between Cp and Cpk?
Cp only compares the tolerance width with the spread; Cpk also considers centring. A process with Cp = 1.39 but its mean shifted towards one limit can have Cpk = 1.11.
What does Cpk = 1 mean?
The nearer specification limit is exactly 3σ from the mean, so about 1,350 ppm fall outside that limit if the data are normal.
What are Cpm and Cpkm?
Capability indices that penalise deviation from a target value T. Cpm replaces σ by √(σ² + (μ − T)²); Cpkm divides Cpk by √(1 + ((μ − T)/σ)²).
What is the difference between Cpk and Ppk?
Cpk uses the short-term (within-subgroup) standard deviation; Ppk uses the overall long-term standard deviation. Enter the SD that matches the index you want.
Does Cpk assume normal data?
The index itself does not, but converting it to ppm or yield does. For skewed data use a transformation or a non-normal capability method.
How many parts do I need for a capability study?
At least 30, and preferably 50–100 measurements from a stable process; with few parts the estimate of σ – and therefore Cpk – is very uncertain.