Process Capability Index - Cp, Cpk, Pp, and Ppk from USL and LSL
Use this process capability index calculator to compute Cp, Cpk, Pp, Ppk, Cpm, sigma level, and PPM defects from USL, LSL, mean, and standard deviation.
Process Capability Index
Results
What Is the Process Capability Index?
A process capability index turns a specification window (USL and LSL), a process mean, and a standard deviation into dimensionless numbers - Cp, Cpk, Pp, Ppk, Cpm - that summarise how well a manufacturing process fits the engineering limits, plus the sigma level and the expected defect rate in parts per million.
Cp measures the width comparison when the process is centered, and Cpk adjusts that comparison for the offset of the mean from the midpoint. Pp and Ppk play the same role using the long-term sample standard deviation, which makes them the indices a production report quotes. The Taguchi Cpm index adds a penalty for squared deviation of the mean from the target, rewarding processes that hit the nominal.
If you do not yet have a sigma, the Standard Deviation Calculator produces the sample or population standard deviation from a comma-separated dataset.
How the Calculator Works
Enter your specification window, process mean, and the two standard deviations. The form applies the ASQ and ISO 22514-2 formulas for Cp, Cpk, Pp, Ppk, and Cpm in one pass, and derives the short-term sigma level (3 * Cpk) plus the expected PPM defect rate from the standard normal distribution.
- USL: Highest acceptable measurement value.
- LSL: Lowest acceptable measurement value.
- mean (x-bar): Arithmetic mean of the measurements, in the same unit as the spec limits.
- sigma: Short-term within-subgroup standard deviation used for Cp and Cpk.
- s: Long-term overall sample standard deviation used for Pp and Ppk.
- target (T): Nominal value used by the Taguchi Cpm index.
PPM and yield use the standard normal tail approximation P(outside) = 2 * (1 - normalCDF(3 * Cpk)). For an off-center process the true defect rate is the sum of the two one-sided tails, so the form reports the conservative two-sided centered number.
Centered bearing diameter process
USL = 10.5 mm, LSL = 9.5 mm, mean = 10.02 mm, short-term sigma = 0.10 mm, overall s = 0.12 mm, target = 10.0 mm, n = 30.
Cpu = 1.60, Cpl = 1.73, Cpk = min(1.60, 1.73) = 1.60, Cp = 1.67, Pp = 1.39, Ppk = 1.33, Cpm = 1.63.
Result: Cp = 1.67, Cpk = 1.60, Pp = 1.39, Ppk = 1.33, sigma level = 4.8.
A Cpk above 1.33 qualifies the process as capable for most ASQ and ISO 22514 customers. The Ppk < Cpk gap flags extra long-term variability, often from tool wear or incoming-material variation.
According to American Society for Quality (ASQ), Cp is (USL - LSL) / (6 sigma), Cpk is min((USL - mean), (mean - LSL)) / (3 sigma), and Pp and Ppk use the overall sample standard deviation.
For the equivalent Z value behind the Cpk-to-PPM conversion, the Z-Score Calculator returns the same number of standard deviations between a value and the mean.
Key Concepts Explained
Four ideas cover most process capability index questions in a Six Sigma Green Belt class.
Cp versus Cpk
Cp compares the specification window to the natural spread and ignores centering. Cpk applies the same comparison to the smaller of the two one-sided gaps between the mean and the spec limits, so they are equal only when the mean sits at the midpoint.
Pp and Ppk (overall performance)
Pp and Ppk replace the short-term sigma with the long-term sample standard deviation. They show up on monthly production reports because they capture tool wear, supplier drift, and shift-to-shift variation.
Cpm (Taguchi capability)
The Taguchi Cpm index adds (mean - target)^2 to the variance inside the square root. A process that hugs the target wins over one that drifts but stays inside the spec window, which is why Cpm is the number to quote when the customer cares about nominal value.
Sigma level and PPM conversion
The short-term sigma level is 3 * Cpk, and the expected two-sided defect rate in parts per million is 2 * (1 - normalCDF(sigma level)). Cpk = 1.00 corresponds to a 3 sigma process and about 2,700 ppm defects; Cpk = 1.33 corresponds to a 4 sigma process and about 63 ppm.
These four concepts cover the language a quality engineer uses when reporting capability to a regulator or a customer.
When the study needs a confidence interval on the mean, the Confidence Interval Calculator returns the lower and upper bounds from the same dataset.
How to Use This Calculator
Six steps take you from a specification drawing to a full capability report, and the same flow works for any capability study.
- 1 Enter the specification limits: Type the USL and LSL from the engineering drawing or quality plan. Use the same unit for every input.
- 2 Add the target value: Type the nominal target (for example 10.0 mm) so the form can report the Taguchi Cpm index. If your specification has no separate nominal, leave the default.
- 3 Enter the process mean and standard deviations: Type the mean, the short-term sigma, and the overall sample standard deviation. The two can differ - that gap is what makes Pp and Ppk interesting.
- 4 Enter the sample size: Type the number of measurements that produced the sample standard deviation. Minimum is 2; a typical study uses 25 to 30 subgroups.
- 5 Read the indices: Cp and Cpk for short-term capability, Pp and Ppk for long-term performance, Cpm if you supplied a target. Compare Cpk to ASQ cut-offs (1.00 marginal, 1.33 capable, 1.67 excellent).
- 6 Compare to PPM and sigma level: Use the sigma level and PPM defect rate to communicate the result in units operations and finance teams recognise.
Practical example: a bearing inner diameter specified at 10.0 +/- 0.5 mm, with 30 measurements producing a mean of 10.02 mm, short-term sigma 0.10 mm, and overall standard deviation 0.12 mm. Enter USL = 10.5, LSL = 9.5, target = 10.0, mean = 10.02, sigma = 0.10, s = 0.12, n = 30. The form returns Cpk = 1.60, Ppk = 1.33, and about 2 ppm defects - a capable process that still needs tool-wear monitoring.
If you would rather quote the process precision as a percentage of the mean, the Relative Standard Deviation Calculator converts the same sample standard deviation into a coefficient of variation that drops straight into the report.
Benefits of Using This Calculator
A capability report is only useful if the numbers inside it are consistent.
One tool for short-term and long-term capability keeps the report consistent.
To compare the measured process mean to the engineering target, the Percent Error Calculator turns the same mean and target into a percent-error number that pairs naturally with the Cpm index.
Factors That Affect Your Results
Five factors shape the numbers you report, plus two limitations worth knowing before you defend them.
Centering of the mean
Cpk equals Cp only when the mean sits at the midpoint. A small shift in mean can drop the result from 1.33 to 1.00 without changing sigma, which is why a capable process needs ongoing mean control.
Choice of sigma
Short-term within-subgroup sigma drives Cp and Cpk; long-term sample standard deviation drives Pp and Ppk. The gap between them signals tool wear, material drift, or shift-to-shift variation.
Sample size
A capability study needs at least 25 to 30 subgroups to give a stable sigma. Smaller samples inflate the apparent index by underestimating the long tail of the distribution.
Distribution shape
The Cp, Cpk, and PPM formulas assume an approximately normal distribution. A skewed or bimodal process can quote Cpk = 1.33 and still produce far more defects than predicted - check a histogram first.
One-sided specifications
Strength, purity, and surface-roughness specs often quote only an upper or a lower limit. The form reports the relevant one-sided index (Cpu or Cpl) instead of forcing a Cp that has no meaning.
- • Cpk, Pp, and Ppk assume the process is in statistical control. Run a control chart first; an out-of-control process inflates sigma and hides the true capability.
- • The PPM conversion uses the standard normal tail approximation P(outside) = 2 * (1 - normalCDF(3 * Cpk)), assuming the process is centered on the worse side. For an off-center process the true defect rate is the sum of the upper and lower tails at Z = Cpu * 3 and Z = Cpl * 3.
In a real quality report the capability section is paired with a control chart and a normality check - the indices are a summary, not the whole story.
According to ISO 22514-2:2017, process capability is the statistical property of a process in statistical control to produce output within specification limits.
According to Wikipedia - Process Capability Index, Taguchi Cpm adds (mean - target)^2 to the variance, and the short-term sigma level equals 3 * Cpk.
When the report also needs to defend the measurement uncertainty behind the sigma, the Absolute Uncertainty Calculator turns the dataset half-range or instrument resolution into the standard uncertainty u that should be quoted alongside Cpk.
Frequently Asked Questions
Q: What is a good Cpk value?
A: ASQ and most Six Sigma textbooks treat Cpk of 1.00 as the minimum acceptable for an existing process, 1.33 as capable, 1.67 as excellent, and 2.00 as world-class. Use 1.33 as the working target unless a customer contract specifies a higher cut-off.
Q: What is the difference between Cp and Cpk?
A: Cp compares the specification window to the natural process spread and ignores centering. Cpk applies the same comparison to the smaller of the two one-sided gaps between the mean and the spec limits. Cpk equals Cp only when the mean sits exactly at the midpoint of the specification window.
Q: How is Cpk calculated from USL, LSL, mean, and standard deviation?
A: First compute the upper one-sided index Cpu = (USL - mean) / (3 * sigma) and the lower one-sided index Cpl = (mean - LSL) / (3 * sigma). Cpk is the smaller of the two, so Cpk = min(Cpu, Cpl). Cp is the width comparison (USL - LSL) / (6 * sigma), and Ppk uses the overall sample standard deviation instead of short-term sigma.
Q: What is the difference between Cpk and Ppk?
A: Cpk uses the short-term within-subgroup sigma and represents the best the process can do. Ppk uses the overall sample standard deviation and represents what the process actually delivers on a long-term production run. A large gap between Cpk and Ppk is the signal that tool wear, supplier drift, or shift-to-shift variation is hurting long-term performance.
Q: How do you convert Cpk to PPM defects?
A: Take the short-term sigma level Z = 3 * Cpk and compute the two-sided normal tail probability P(outside) = 2 * (1 - normalCDF(Z)). Multiply by one million to get the expected defect rate in parts per million. A Cpk of 1.00 corresponds to Z = 3 and about 2,700 ppm; a Cpk of 1.33 corresponds to Z = 4 and about 63 ppm; a Cpk of 1.67 corresponds to Z = 5 and about 0.57 ppm.
Q: What does a Cpk of 1.33 mean?
A: A Cpk of 1.33 means the shorter one-sided gap between the mean and the nearest specification limit is 1.33 standard deviations away from the mean - about 4 sigma total. The expected defect rate is roughly 63 parts per million, which most ASQ and IATF 16949 customers treat as the working threshold for a capable process.