Process capability calculator, Cp and Cpk
Your data
The deviation has to come from a process that is already stable. On a process that is still jumping around, these indices describe a thing that does not hold still long enough to be described, and the number will look fine right up until the day it does not.
The figures below assume the measurements follow a bell curve. That assumption is what turns an index into a defect rate, and it is the first thing to check when the rate you predict and the rate you see disagree.
Results
Capability once the position of the process is counted, not only its spread
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| Capability counting the spread alone | — |
| Room left towards the upper limit | — |
| Room left towards the lower limit | — |
| Parts out of specification, per million | — |
| How far off the middle the process sits, in standard deviations | — |
| Parts per million it would make if it were centred, with no other change | — |
| Cut in the deviation that would be needed to match that without centring, in percent | — |
What drifting off centre does, with the variation left untouched
| Standard deviations off the middle | Capability counting the spread alone | Capability counting position too | Parts out of specification, per million |
|---|
The second column never moves down that table. Nothing about the variation changed; the process only moved sideways. Yet the defect rate climbs by orders of magnitude, because what falls outside a limit depends on how far the process sits from it, and a width-only index cannot see that at all.
This is also where the famous figure for six sigma quality comes from. A process whose specification is six standard deviations wide on each side makes about two defects in a thousand million when it is centred. The number everyone quotes instead is three point four per million, and the whole of that difference is an allowance for the process wandering one and a half standard deviations off centre over the long run. Same variation, same limits, seventeen hundred times the defects.
The practical reading is that centring is usually the cheap half of the work and gets done last. Moving where a process sits is an adjustment; making it vary less is a project. When the two indices disagree, the gap between them is telling you how much is available for the price of the adjustment.
What is the difference between Cp and Cpk?
Cp compares the width of the specification with the width of the process and nothing else. Cpk compares the distance from the process to each limit, so it sees both the width and where the process is sitting.
That is why they can disagree wildly. A process whose specification is twelve standard deviations wide scores Cp = 2.00 no matter where it sits. Centred, its Cpk is also 2.00 and it makes 0.002 defects per million. Sitting halfway to a limit, the Cp is still 2.00, the Cpk falls to 1.00 and it makes 1350. With the average right on the limit, Cp is still 2.00, Cpk is zero and half of everything is out of specification.
| Where the process sits | Cp, width only | Cpk, width and position | Defects per million |
|---|---|---|---|
| Centred | 2.00 | 2.00 | 0.002 |
| Halfway to the limit | 2.00 | 1.00 | 1350 |
| On the limit | 2.00 | 0.00 | 500000 |
Why is six sigma quality 3.4 defects per million and not much less?
Because the famous number already contains an allowance for the process drifting off centre. A specification six standard deviations wide on each side, with the process centred, gives about 0.002 defects per million, not 3.4.
The 3.4 figure is that same process shifted one and a half standard deviations off centre, which is the drift the original Motorola work assumed a real process shows over the long run. Nothing about the variation changed between the two numbers; only the centring did, and it costs a factor of about seventeen hundred.
| Standard deviations off centre | Cp, width only | Cpk, width and position | Defects per million |
|---|---|---|---|
| 0.0 | 2.00 | 2.00 | 0.002 |
| 0.5 | 2.00 | 1.83 | 0.019 |
| 1.0 | 2.00 | 1.67 | 0.287 |
| 1.5 | 2.00 | 1.50 | 3.4 |
| 2.0 | 2.00 | 1.33 | 31.7 |
Should I reduce variation or centre the process first?
Centre it first, almost always, because it is the cheaper of the two and the calculator will tell you exactly what it is worth. Moving where a process sits is usually an adjustment; making it vary less is usually a project.
A process with Cp = 1.33 sitting one standard deviation off centre makes 1350 defects per million. Centring alone takes it to 63. To reach that same 63 without centring you would have to cut the standard deviation by nearly twenty two per cent, which is a different order of effort for the same result.
What Cpk is good enough?
1.33 is the usual floor for an established process and 1.67 is common where a failure is expensive or hard to detect later. Below 1.00 the specification is narrower than the process can hold, and screening every part becomes the only way to keep bad ones in.
Treat those numbers as conventions rather than laws. What they really encode is a defect rate you are willing to live with, and the rate is the thing to argue about: a centred process at 1.00 makes 2700 parts per million, at 1.33 makes 63 and at 1.67 makes 0.6.
| Cpk, process centred | Defects per million |
|---|---|
| 1.00 | 2700 |
| 1.33 | 63 |
| 1.67 | 0.6 |
| 2.00 | 0.002 |
Can Cpk be negative?
Yes, and it means the average has landed outside the specification limits, so more than half of what the process makes is already outside them.
It is not a capability problem to be improved by degrees at that point. It is a setting to be corrected, and the indices only start describing anything useful once the process is back inside its limits.
Why does my calculated defect rate not match what I actually see?
The commonest reason is that turning an index into a defect rate assumes the measurements follow a bell curve, and many real processes do not. Anything with a hard physical floor, or a tool that wears in one direction, produces a lopsided distribution, and the tail you care about is exactly where the assumption fails hardest.
The second reason is stability. These indices describe a process that holds still. If yours drifts between shifts or between batches, the deviation you measured over a short window is not the deviation you live with, and the rate will look better on paper than in the crate.
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