What Is Cpk? How to Calculate Cp and Cpk for CNC Machining

Many machine shops have been here: every part is within tolerance, yet the customer says the Cpk is not good enough. So what is Cpk, and why can a process with zero rejects still be judged as not capable? This article explains Cp and Cpk, walks through a calculation using a turned shaft, covers the usual reasons Cpk drops on CNC lathes, and shows how to use offset compensation in a controlled way to keep it stable.

What Is Cpk and How Is It Different from Cp?

Cp and Cpk are process capability indices. They tell you whether your process variation is narrow enough compared with the tolerance on the drawing.

• Cp compares the tolerance width with the spread of the process (six standard deviations). It ignores where the average sits, so it describes the process's potential.
• Cpk also accounts for the location of the mean, measuring the distance from the mean to the nearest specification limit. It describes what the process is actually delivering.

High Cp with low Cpk means the machine's spread is fine but the average has shifted towards one limit. On a CNC machine this is often fixed by adjusting the offset to re-centre the process. If Cp itself is low, the problem is variation, and offsets alone will not solve it.

How to Calculate Cp and Cpk

The basic formulas

• Cp = (USL − LSL) ÷ (6σ)
• Cpu = (USL − μ) ÷ (3σ)
• Cpl = (μ − LSL) ÷ (3σ)
• Cpk = the smaller of Cpu and Cpl

USL and LSL are the upper and lower specification limits, μ is the process mean and σ is the process standard deviation.

Hypothetical example: shaft diameter

Say the drawing calls for 20.000 ±0.010 mm (LSL 19.990, USL 20.010). Your data show a mean of 20.004 mm and σ of 0.002 mm.

• Cp = 0.020 ÷ 0.012 ≈ 1.67
• Cpu = (20.010 − 20.004) ÷ 0.006 = 1.00
• Cpl = (20.004 − 19.990) ÷ 0.006 ≈ 2.33
• Cpk = 1.00

The machine has good potential, but because the mean sits 0.004 mm above nominal, Cpk falls to 1.00. Bring the mean back to around 20.000 mm with an offset change and Cpk moves close to Cp.

Acceptance levels vary by customer. In automotive supply chains a minimum Cpk of 1.33 is a common requirement, with higher values often requested for critical characteristics or new processes. Always confirm the exact requirement with your customer.

Cp/Cpk vs Pp/Ppk, and What to Check First

The formulas look the same; the difference is how σ is estimated.

• Cp/Cpk use within-subgroup variation, for example estimated from R-bar or S-bar of consecutive parts. This reflects short-term variation.
• Pp/Ppk use the overall standard deviation of all the data, so tool wear, shift changes, temperature and material lots are all included.

On CNC work, a good Cpk with a much lower Ppk usually points to the mean drifting over time, typically from tool wear or thermal growth.

Before calculating, make sure that:

1. The process is statistically stable, with no out-of-control points or unusual trends on the control chart.
2. The data are reasonably close to a normal distribution, or you use a method suited to the actual distribution.
3. You have enough data; small samples give unreliable indices.
4. Your measurement system is trustworthy and has passed a Gage R&R study, because gauge variation is counted as process variation.

Why Cpk Drops on CNC Turning

A shifted mean (offsets can fix it)

• Gradual size change from insert wear
• Offsets not re-centred after an insert change
• Thermal drift at start-up or after a long stop
• Operators deliberately running towards one side of the tolerance

Excess variation (fix the root cause)

• Inconsistent workholding; see our guide to jig and fixture design and the 3-2-1 principle
• Play in the spindle or turret
• Variation in raw material hardness or stock size
• Unsuitable cutting conditions causing chatter
• A measurement system with too much variation

A common mistake is calculating Cpk from data that mixes several machines, spindles or fixtures. You cannot see where the variation comes from, so always separate data by source first.

Using Offsets to Raise Cpk Without Over-Adjusting

When the main issue is a shifted or drifting mean, tool and wear offset compensation is the right tool, but it needs rules. Adjusting after every single measurement without filtering noise (over-adjustment, or tampering) increases variation rather than reducing it.

• Use a dead band: if the deviation is smaller than a set threshold, leave it alone.
• Decide on averages: act on a trend across several parts, not one reading.
• Limit the step size: cap each correction to avoid overreacting to a bad measurement.
• Log every change: keep a record of how much was adjusted and why.

Having operators read CMM reports and type offsets by hand risks entering values on the wrong machine, the wrong tool number or with the wrong sign, and results depend on individual experience. An auto offset system applies the same correction rules on every shift. For the principles, read What Is Auto Offset for FANUC CNC.

NKE developed NKE AOC, a centralised system that takes results from CMMs and writes tool and wear offsets to FANUC CNC machines automatically. Consistent re-centring of the mean is one of the key factors in keeping Cpk close to the process's Cp. Note that compensation does not reduce short-term variation from fixtures or the machine itself; that still has to be addressed at the source.

FAQ

Can Cpk be negative, and what does that mean?

Yes. If the process mean lies outside one of the specification limits, Cpk becomes negative, meaning most parts are out of tolerance. Stop and check the set-up immediately.

How many parts do I need to calculate Cpk?

It depends on your customer's requirements and the reference manual you follow. The principle is to collect enough data that reflects real production conditions; too few parts give an unreliable result. Agree the sample size with your customer before you start.

If Cpk is low, where should I start?

Compare Cp with Cpk. If Cp is good but Cpk is low, work on centring and offset compensation. If Cp is also low, check the measurement system, workholding, machine condition and cutting conditions first.

If your Cpk is falling short, or you want to stop entering offsets by hand from CMM reports, NKE's engineers are happy to review your process and suggest a practical way forward. Contact NKE, call 02-746-7196 or reach us on LINE @nke.th.