Process capability calculator (Cp, Cpk)
Cpk is the distance from the mean to the nearer limit in units of 3σ: Cpk = min(USL − x̄, x̄ − LSL) / 3σ. A process with limits 8 and 20, a mean of 16 and σ of 2 has Cp 1.00 but Cpk 0.67: it is wide enough, but off centre, and about 2.3 % of its parts land above the upper limit. Enter a mean and σ, or paste measurements, for Cp, Cpk, Pp, Ppk and the expected ppm out of each limit.
- Cpk
- Cp
- Ppk
- –
- Pp
- –
- Expected out of spec
- Above USL
- 22750ppm
- Below LSL
- 31.7ppm
- Z to USL
- 2.00
- Z to LSL
- 4.00
ppm and Z from σ within.
Meaningful only for a stable process with roughly normal data: check a control chart and the histogram first.
How it’s calculated
Cp = (USL − LSL)/6σ and Cpk = min(USL − x̄, x̄ − LSL)/3σ, with σ within: the moving range of neighbouring values over d₂ = 1.128 for individuals, or the mean subgroup range over d₂, or the mean subgroup standard deviation over c₄. Pp and Ppk are the same with s, the standard deviation of every value. With one limit only the one-sided index is given. The expected ppm is a million times the normal tail beyond Z = (USL − x̄)/σ or (x̄ − LSL)/σ, using s when it is known.
Subgroup constants
| Subgroup size | d₂ | c₄ |
|---|---|---|
| 2 | 1.128 | 0.7979 |
| 3 | 1.693 | 0.8862 |
| 4 | 2.059 | 0.9213 |
| 5 | 2.326 | 0.9400 |
| 6 | 2.534 | 0.9515 |
| 7 | 2.704 | 0.9594 |
| 8 | 2.847 | 0.9650 |
| 9 | 2.970 | 0.9693 |
| 10 | 3.078 | 0.9727 |
R̄/d₂ is for subgroups of 10 or fewer; s̄/c₄ takes up to 25. For individuals the moving range of two uses d₂ = 1.128.
When the numbers mean something
Capability compares a stable process with its limits. If the process drifts, shifts or jumps, Cp and Cpk describe nothing that will repeat: put the data on a control chart first. Cpk well above Ppk is itself a sign of drift between subgroups.
The indices and the ppm assume normal data. Far tails are where real processes differ most from the model, so a ppm of 0.5 is an estimate of the model, not a count of parts. And they are estimates from a sample: around 50 values is the usual minimum, and 100 or more is better.
Cpk or Ppk?
Both use the same formula. Cpk takes σ within, the spread over a short run (inside subgroups, or between neighbouring parts), so it says what the process can do. Ppk takes the standard deviation of every value, so drift and shifts between subgroups count against it, and it says what the parts actually did. A Ppk well below Cpk means the process moves about more than its short-term spread suggests.
Common targets
A centred process at Cp 1.00 puts about 2700 ppm outside the limits; at 1.33 about 66 ppm; at 1.67 about 0.5 ppm. Customer requirements often name 1.33 or 1.67, sometimes different values for a new process and for ongoing production. Those are contract numbers: check what your customer’s quality clauses require.