The standard Cp/Cpk/Pp/Ppk formulas assume two things that don't always hold: a two-sided specification with both an upper and lower limit, and an underlying normal distribution. A meaningful share of real characteristics violate one or both assumptions — flatness, roundness, concentricity, and anything bounded at zero are common examples — and running the standard formulas on that kind of data without adjustment produces indices that misrepresent actual capability, sometimes badly.
One-Sided Specifications
Plenty of characteristics only have a single meaningful limit. A minimum wall thickness has a lower spec limit but no meaningful upper one — thicker is fine. A maximum contamination level has an upper limit but no lower one — cleaner is always acceptable. Running the two-sided Cpk formula on this kind of characteristic doesn't work, because there's no second limit to calculate a distance to.
The correct approach uses a one-sided capability index calculated against whichever single limit actually applies — the same underlying logic as Cpk (distance from the mean to the limit, divided by three sigma) but without averaging against a nonexistent second boundary. Reporting a standard two-sided Cpk on a one-sided characteristic by inventing or ignoring the missing limit produces a number that either overstates or understates real capability, depending on which direction the omission goes.
The distinction matters practically because it changes how the process should be centered. A two-sided characteristic benefits from centering the mean between both limits. A one-sided characteristic benefits from pushing the mean as far from the single limit as economically sensible in the safe direction — there's no such thing as "too far" from a lower spec limit on wall thickness, within reason, the way there would be with a spec that has real limits on both sides.
Non-Normal Distributions
The standard capability formulas assume the underlying data follows a normal distribution, and that assumption is what makes the three-sigma-based math meaningful in the first place. Several common characteristics are inherently non-normal: anything physically bounded at zero (concentricity, flatness, most measures of "how far off" something is from ideal) tends to produce a right-skewed distribution, since values can't go below zero but can extend arbitrarily far above it.
Running standard Cpk math on skewed data produces a systematically misleading index, because the three-sigma assumption behind the formula doesn't match how the data actually spreads. A skewed distribution can have a substantial fraction of its data closer to one limit than the normal-distribution math accounts for, meaning the calculated Cpk can look comfortable while the actual proportion of parts near or beyond the limit is considerably higher than the index suggests — or the reverse, understating a process that's actually performing fine.
Two Ways to Handle Non-Normal Data
Transformation applies a mathematical function — Box-Cox or Johnson transformation are the standard choices — to convert the non-normal data into something close enough to normal that the standard Cpk formulas become valid again, calculates capability on the transformed data, then reports the result. This works well when a transformation genuinely normalizes the distribution, but it adds a layer of abstraction between the reported index and the raw data that needs to be documented clearly — a Cpk calculated on Box-Cox transformed data isn't directly comparable to a Cpk calculated on raw data from a different, normally distributed characteristic.
Distribution-specific formulas skip the transformation step and instead fit the actual underlying distribution — Weibull is common for life-data and certain bounded characteristics — and calculate capability indices using percentiles from that distribution directly rather than assuming normality. This avoids the interpretability cost of a transformed scale but requires correctly identifying which distribution actually fits the data, which itself takes a distribution-fit check rather than an assumption.
The Step That Gets Skipped
The mistake underlying both failure modes — one-sided specs treated as two-sided, non-normal data run through normal-distribution formulas — is the same: capability calculations run without first checking what kind of data is actually being fed into them. A normality check and a review of which spec limits genuinely apply should precede the capability calculation, not follow it as a footnote after a suspicious-looking number prompts someone to look closer.
Calculating Capability Correctly for the Data You Have
SigmaDesk's process capability calculator includes a built-in normality check as a standard step before capability indices are calculated, flagging when standard assumptions don't hold — free in the browser, part of the full SigmaDesk SPC platform alongside control charts and Gage R&R.
A capability index calculated against the wrong assumptions isn't a rough approximation of the truth. It's a precise answer to a question that doesn't match the actual process.