Cp vs Cpk: Why Your Process Can Be Capable on Paper and Still Make Scrap

Your capability report shows C_p = 1.67 — comfortably above the 1.33 your customer asks for. The same study reports C_pk = 1.00, and parts are drifting past the upper spec limit on the floor. Nothing is wrong with the arithmetic. C_p and C_pk answer two different questions, and the gap between them points straight at where your scrap is coming from.

This post walks the Cp vs Cpk difference with a worked example, the centering math behind it, and a clear verdict on which index to report to whom. The short version: C_p tells you what the process could do if it were centered; C_pk tells you what it is actually doing right now.

The one-line difference

C_p compares the width of your specification to the spread of your process and assumes the process sits dead center between the limits. It is a measure of potential — the best capability the spread alone would allow.

C_pk drops the centering assumption. It measures the distance from the process mean to the nearer specification limit, so any drift off nominal pulls the number down. C_pk is always less than or equal to C_p, and the two are equal only when the process is perfectly centered.

Tip Read the pair together, never C_p alone. A high C_p with a much lower C_pk is not a contradiction — it is a centered-spread problem hiding inside a consistent process.

The formulas, and what each one ignores

Both indices use the within-subgroup standard deviation (the short-term sigma you estimate from your control chart), so they describe the same process spread. Only the centering treatment changes.

Key Formulas $$C_p = \frac{USL - LSL}{6\sigma}$$ $$C_{pk} = \min\left(\frac{USL - \mu}{3\sigma},\; \frac{\mu - LSL}{3\sigma}\right)$$

The C_p formula contains no process mean at all — it cannot see where the distribution sits between the limits. C_pk inserts the mean twice and keeps the worse of the two sides. That single change is the entire story.

You can also write C_pk in terms of C_p and a centering factor, which makes the penalty explicit:

Centering Factor $$C_{pk} = (1 - k)\,C_p, \qquad k = \frac{|\mu - M|}{(USL - LSL)/2}, \qquad M = \frac{USL + LSL}{2}$$

Here M is the midpoint of the tolerance and k runs from 0 (perfectly centered) toward 1 (mean sitting on a spec limit). When k = 0, C_pk equals C_p. Every point of off-centering costs you capability you already paid for in tight variation.

A worked example where Cp passes and Cpk fails

Take a turned diameter with USL = 10.5 mm and LSL = 9.5 mm, so the tolerance width is 1.0 mm and the midpoint M is 10.0 mm. The process is tight but running high: mean μ = 10.2 mm, within-subgroup sigma = 0.10 mm.

Worked Example C_p = (10.5 − 9.5) / (6 × 0.10) = 1.0 / 0.60 = 1.67
C_pk = min[ (10.5 − 10.2) / (3 × 0.10), (10.2 − 9.5) / (3 × 0.10) ]
= min[ 0.30 / 0.30, 0.70 / 0.30 ] = min[ 1.00, 2.33 ] = 1.00
Check with the centering factor: k = |10.2 − 10.0| / 0.50 = 0.40, so C_pk = (1 − 0.40) × 1.67 = 1.00. The two methods agree.

The spread is excellent — a C_p of 1.67 would clear an automotive critical-characteristic requirement. But the mean sits 0.2 mm high, only 0.3 mm from the upper limit, so C_pk collapses to 1.00 and the process is not capable by the usual 1.33 bar.

Now recenter the mean to nominal (10.0 mm) without touching the variation. Both sides become 0.5 mm / 0.30 mm = 1.67, so C_pk rises from 1.00 to 1.67. You bought a capable process by moving a tool offset.

Tip Recentering a mean is almost always cheaper and faster than reducing variation. When the C_p − C_pk gap is large, take the centering win first — adjust the offset, fixturing, or setpoint — before you spend money on a tighter machine.

What each index actually tells you

Question	C_p	C_pk
Accounts for process centering?	No	Yes
What it represents	Potential capability if centered	Actual capability as the process runs
Uses the process mean?	No	Yes (penalizes the nearer limit)
Relationship	C_pk ≤ C_p; equal only when perfectly centered
Best used for	Machine studies, tolerance design, ceiling on improvement	Customer reporting, PPAP, ongoing capability

Common Mistake Reporting C_p by itself on a capability summary. A reader sees 1.67 and signs off, never learning the process is making out-of-spec parts because the mean is off nominal. If you report one number to a customer, report C_pk.

The thresholds that matter — and where Ppk comes in

The conventional capability bar is C_pk ≥ 1.33 for a capable process, with C_pk ≥ 1.67 expected for critical characteristics in automotive and aerospace work. IATF 16949:2016 requires capability reporting for designated key characteristics, and the AIAG SPC Reference Manual is the methodology most North American suppliers cite for how those indices are calculated.

C_p and C_pk both use the short-term, within-subgroup sigma, so they describe a snapshot of a stable process. Their long-term cousins P_p and P_pk use the overall standard deviation across the whole study. When P_pk trails C_pk, the process is drifting between subgroups — that gap is a separate diagnostic from the C_p − C_pk centering gap. We cover it in detail in why the gap between Cpk and Ppk signals process instability.

One caution before you trust any of these numbers: a capability index computed from too few parts is noisy. See how many parts you need before a Cpk number is trustworthy for the sample-size math, and the histogram patterns a single Cpk value can hide when the distribution is not normal.

Which index to report — verdict by role

Process or manufacturing engineer evaluating a machine: lead with C_p. You want the ceiling the equipment can reach independent of today's setup, then use the C_p − C_pk gap to decide whether to recenter or reduce spread.
Quality engineer preparing a customer or PPAP submission: report C_pk (and P_pk where the customer asks for long-term performance). C_pk is the honest answer to “can this process meet the print as it runs?”
Quality manager reviewing a dashboard: watch the C_p − C_pk gap across characteristics. A wide gap flags a centering problem you can fix cheaply; a low C_p flags a variation problem that needs real investment.

If you are building these indices from measurement data, our control chart and capability tool calculates C_p, C_pk, and the within-subgroup sigma from an uploaded data set, so you can see the centering gap without rebuilding the formulas in a spreadsheet.

For the underlying methodology, the ASQ process capability resource and the NIST/SEMATECH e-Handbook section on process capability both lay out the C_p and C_pk definitions in full.