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Manufacturing & Production Engineering DISCUSSION
We turn a shaft diameter of 25.00 ± 0.05 mm. From 50 measured parts the mean is 25.022 mm and the standard deviation is 0.010 mm. The spreadsheet reports Cp = 1.67 and Cpk = 0.93, and the customer requires a Cpk of at least 1.33.
Why are the two indices so different when they come from the same data, how many parts out of tolerance does this mean, and what do I actually need to change on the machine?
Cp compares the tolerance width with the process spread and ignores where the process sits: Cp = (USL - LSL) / 6σ = 0.10 / 0.060 = 1.67. Cpk uses the distance from the mean to the nearer limit: Cpk = min(USL - mean, mean - LSL) / 3σ. The upper side gives (25.050 - 25.022) / 0.030 = 0.93 and the lower side (25.022 - 24.950) / 0.030 = 2.40, so Cpk = 0.93.
Read together, they say the spread is small enough but the process is running 0.022 mm high. Cp is what you could achieve if it were centred; Cpk is what you are getting.
For the defect rate, convert the distance to the limit into standard deviations. The upper limit is (25.050 - 25.022) / 0.010 = 2.8σ from the mean, and a normal distribution has about 0.26 percent of its area beyond 2.8σ, so roughly 2,600 parts per million will be oversize. The lower limit is 7.2σ away and contributes nothing measurable.
Centred at 25.000 mm with the same σ, both limits would be 5σ away and the predicted total would be well under 1 ppm. That is the gap between your two indices expressed in parts.
The fix is an offset, not a better machine. Shift the tool wear offset by about 0.022 mm on diameter so the mean lands on 25.000, and Cpk rises to the Cp value of 1.67 with no change in variation. Then find out why it drifted: on a lathe the usual causes are insert wear, thermal growth during warm-up and an offset that was set from a single part.
An X-bar and R chart with small subgroups taken at regular intervals will show drift early and tell the operator when to adjust. That is better than correcting after every measurement, which adds variation.
Two cautions about the numbers themselves. With 50 parts a capability index has a wide confidence interval, roughly ±0.3 at 95 percent confidence for an index near 1.67, so one good sample is not proof that you will hold 1.33 every time. And the calculation assumes a stable, roughly normal process. If the 50 parts include a tool change or a warm-up drift, the overall standard deviation is larger than the short-term one; this is the difference between Ppk (overall σ) and Cpk (within-subgroup σ), and customers often ask for both.
Plot a histogram and a run chart before quoting any index.
Make sure the gauge is not part of the problem. With a total tolerance of 0.10 mm, the measuring system should consume no more than about 10 percent of it, so repeatability and reproducibility of around 0.01 mm in total. A hand micrometer with 0.01 mm graduations used by several operators can easily exceed that, and its variation is added to the real process variation in your σ.
A short gauge R&R study (for example 10 parts, 3 operators, 2 or 3 readings each) separates the two. It is not unusual for capability to improve simply by moving to a better gauge or a fixed measuring routine.
