Process Capability Calculator (Cp, Cpk, Pp, Ppk)

The tool estimates the spread of the process in two ways and compares each with the specification. With n readings x₁, x₂, …, x_n in collection order and their mean x̄:

1. The moving ranges are the absolute differences between consecutive readings, MR_i = |x_i − x_i−1|, for i = 2 to n. Their average MR̄ measures short-term, reading-to-reading variation.
2. The within-process standard deviation divides MR̄ by d₂ = 1.128, the constant for ranges of two consecutive values.

3. The overall standard deviation is the ordinary sample standard deviation of all the readings, with n − 1 in the denominator. It includes every source of variation in the data, such as shifts and drifts between the start and the end of the run.

4. Cp and Pp compare the width of the specification with six standard deviations. They need both limits and ignore where the mean sits.
5. Cpk and Ppk compare the distance from the mean to the nearer limit with three standard deviations: the smaller of the upper and lower one-sided indices. With only one limit, that side alone gives Cpk and Ppk.

The guide example, ten thickness readings with LSL 9.50 µm and USL 10.50 µm, gives σ_within = 0.0394 µm and s = 0.1076 µm, so Cpk = 3.113 and Ppk = 1.140, with Cp = 4.230. The details under the result list Cp, Pp, MR̄, the minimum and maximum, the four one-sided indices and the distance from the mean to the nearer limit.

• Individual readings only. The within estimate uses moving ranges of single readings. Data collected in rational subgroups, such as five parts every hour, need a subgroup estimate (R̄/d₂ or s̄/c₄), which this tool does not provide.
• Collection order matters. The moving ranges are only meaningful when the readings are pasted in the order they were collected. Sorted or shuffled data give a different, usually smaller, σ_within and a higher Cpk.
• No defect rate. The tool does not convert the indices into ppm or a fraction out of specification. That conversion assumes a stable process and a known distribution shape, which the indices alone do not show.
• Small samples are screening estimates. Both standard deviations are uncertain when there are few readings. Below 25 readings the result says so; a capability qualification usually needs a larger, stable data set defined in your procedure.
• A standard deviation of zero does not mean infinite capability. When all readings are identical, often because the gauge resolution is too coarse, no index is calculated.
• Negative values. A negative Cpk or Ppk means the mean lies outside the specification on that side. The value is shown as calculated, not set to zero.
• For the reasoning behind the two estimates, read the Cpk vs Ppk guide and the I-MR chart guide.

• Reporting Cpk as if it were Ppk. Cpk uses the short-term, within spread and is usually the higher of the two. A customer asking for long-term performance needs Ppk; label which one you report.
• Sorting before calculating. Sorting removes the time sequence and shrinks the moving ranges, so σ_within falls and Cpk rises for the same data. Keep the readings in collection order.
• Using Cpk to claim the process is stable. An index summarises spread and centering; it does not test for shifts or trends. Check a control chart, such as an I-MR chart, first. In the guide example a level shift halfway through still gives Cpk 3.113.
• Treating a small sample as a qualification. Ten or twenty readings give a first estimate. Qualification needs the sample size, period and stability evidence your procedure defines.