Fifty measured values are on the table, the supplier is waiting for sign off, and the specification says Cmk 1.67. Whether the machine goes into series production now depends on one number, and the math behind it fits on a beer mat. Even so, acceptance tests fail on it again and again: the wrong standard deviation, the wrong limit, or a tolerance that was never entered correctly.
Below you will find the Cmk formula, a worked example on a fastening tool and the matching Excel formulas. If you are short on time, paste your measurement series straight into our free online calculator.
01 · Basics
What does the Cmk value tell you?
Cmk is the machine capability index. It shows whether a machine produces a characteristic so tightly and so centrally that the parts stay inside the tolerance with enough margin. The k stands for the location of the mean, so Cmk assesses spread and location together.
It is determined in a machine capability study. The machine produces an uninterrupted run under conditions kept as constant as possible: same material, same operator, no change of settings. Whatever variation remains comes from the machine itself. Our article on the machine capability study describes the procedure and its conditions. This article focuses on the calculation.
Cmk comes with Cm, which measures only the spread and ignores the location. Using both, you can tell a setting problem from a genuine machine problem, as section 06 shows.
02 · Formula
What is the Cmk formula?
You need four inputs: the upper specification limit USL, the lower specification limit LSL, the mean x̄ of the measurement series and its standard deviation s. They give you four indices.
The factor 3 in the denominator comes from the normal distribution: there, 99.73 percent of values lie within three standard deviations of the mean. A Cmk of 1.00 therefore means the nearer limit sits exactly three standard deviations from the mean. At 1.67 it is five standard deviations. That distance is the buffer the machine needs later in series production.
It matters which standard deviation you use. The study uses the sample standard deviation across all values, with n minus 1 in the denominator. Using the population standard deviation by mistake gives a slightly too good Cmk with 50 values. The difference is small, but with tight results it can decide between pass and fail.
The formulas are the same as for Cp and Cpk. The difference lies in the data: for Cpk they come from the running process, for Cmk from the short, controlled run.
03 · Example
Calculating Cmk: a worked example on a fastening tool
An electric nutrunner is to be accepted for a joint with a target torque of 25.0 Nm. For this example we assume a tolerance of 22.5 to 27.5 Nm. The tool tightens 50 joints in a row on a test bench, and a calibrated transducer records every torque value. This is how the calculation works:
The table shows three cases with the same tolerance, each changing only one input. In case 2 the tool is just as precise, but its location has shifted up by 0.6 Nm. In case 3 the mean sits perfectly, but the tool scatters more.
| Case | x̄ in Nm | s in Nm | Cm | Cmu | Cml | Cmk | Result |
|---|---|---|---|---|---|---|---|
| 1 Baseline | 25.30 | 0.38 | 2.19 | 1.93 | 2.46 | 1.93 | capable |
| 2 Shifted mean | 25.90 | 0.38 | 2.19 | 1.40 | 2.98 | 1.40 | not capable |
| 3 Higher spread | 25.00 | 0.55 | 1.52 | 1.52 | 1.52 | 1.52 | not capable |
Cases 2 and 3 both fail, for entirely different reasons. In case 2, Cm stays at 2.19 and only Cmk drops. The tool can work tightly, it just aims off target. In case 3, Cm and Cmk are equally low, because the mean is centred and the spread itself is the problem. Only the comparison of Cm and Cmk shows where the cause lies.
04 · Calculator
Calculate Cmk online with the CSP calculator
CSP Intelligence is a software company from Lower Bavaria, Germany, that captures, analyses and archives quality data in manufacturing. For quick evaluation of measurement series, CSP offers a free online calculator. It runs entirely in your browser, and your data never leaves your device.
The tool is called a Cpk calculator, but it works directly for machine capability. It calculates the sample standard deviation with n minus 1 across all values entered. Cmk is calculated the same way. If you paste in the 50 values from your study, the Cp it shows equals your Cm and the Cpk equals your Cmk.
Calculate Cmk in under a minute
Paste your 50 measured values and get Cm and Cmk instantly, free of charge and without registration.
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05 · Excel
How do you calculate Cmk in Excel?
Many quality departments evaluate their studies in Excel. The calculation needs only a few cells. In the layout below, the 50 values are in cells A2 to A51 and the limits in E1 and E2.
| Cell | Content | Formula |
|---|---|---|
| E1 | USL | 27.5 |
| E2 | LSL | 22.5 |
| E3 | Mean x̄ | =AVERAGE(A2:A51) |
| E4 | Standard deviation s | =STDEV.S(A2:A51) |
| E5 | Cm | =(E1-E2)/(6*E4) |
| E6 | Cmu | =(E1-E3)/(3*E4) |
| E7 | Cml | =(E3-E2)/(3*E4) |
| E8 | Cmk | =MIN(E6,E7) |
Check the function in E4. STDEV.S calculates the sample standard deviation with n minus 1. STDEV.P returns the population standard deviation and makes Cmk slightly too good.
Excel gets difficult once the study is repeated regularly. Each evaluation sits in its own file, and nothing checks that the values really belong to the right tool, date and inspection plan. An auditor will ask exactly that.
06 · Interpretation
Reading Cm and Cmk: setting problem or spread problem?
Cmk alone tells you whether the machine passes. Comparing it with Cm tells you what to do if it fails. This saves time, because the two causes call for completely different actions.
| Finding | Meaning | Typical action |
|---|---|---|
| Cm high, Cmk high | Tight spread, centred | Accept and document |
| Cm high, Cmk low | Tight spread, mean shifted | Correct setting or offset, repeat study |
| Cm low, Cmk low | Machine spread too large | Look for causes in mechanics, drive or tool |
| Cmk greater than Cm | Mathematically impossible | Check formula, limits and data |
The last row is a handy plausibility check. Cmk can never exceed Cm. If an evaluation shows a higher Cmk, the limits are almost always swapped or the formulas point to the wrong cells.
07 · Limits
What Cmk value is required?
There is no single limit that applies to everyone. What counts is what your customer, your internal standard or the relevant guideline specifies. Two thresholds are common, though.
| Cmk | Assessment | Typical use |
|---|---|---|
| below 1.33 | not capable | improve the machine, no release |
| 1.33 to below 1.67 | limited | sometimes accepted in general industry |
| 1.67 and above | capable | frequent requirement, especially in automotive |
Why is the requirement for Cmk higher than the usual 1.33 for Cpk? The study runs under ideal conditions. In series production, material batches, changing operators, tool wear and temperature add to the spread. The higher Cmk creates the buffer so the process remains capable later on.
Measurement system capability is a separate topic with its own indices, but it comes first: before you determine a Cmk, the measuring equipment itself must be capable. A measurement system that scatters too much distorts every Cmk.
08 · Fastening
Cmk in fastening technology: what VDI/VDE 2645 requires
In fastening, Cmk is everyday business. Electric nutrunners, pulse tools and torque wrenches are regularly tested for machine capability. The German guideline series VDI/VDE 2645 describes three levels for this.
The three parts build on each other. Without a capable measuring system, no Cmk is reliable, and without a capable tool, no process capability is achievable. The guideline VDI/VDE 2862 additionally defines which requirements apply to joint categories A, B and C. Part 2 of that guideline explicitly calls for capable fastening tools with a machine capability study for plant and mechanical engineering. Part 1 for the vehicle industry was reissued in September 2026, replacing the 2012 edition.
For a plant this means many tests, because every tool is checked again at fixed intervals and after every repair. CSP QST is the software in the CSP Manufacturing OS for tool and process testing. It schedules these tests, takes the values from the test device and calculates Cm and Cmk for each tool automatically.
09 · Practice
Common mistakes when calculating Cmk
Most wrong Cmk values are not calculation errors in the narrow sense. They come from wrong input data or wrong assumptions. Four mistakes are especially common.
Mistake 3 mainly affects one sided characteristics. Characteristics such as runout, flatness or roughness only have an upper limit, and values cluster near zero. A normal distribution fits poorly. In such cases, a better fitting distribution model is chosen and the capability index derived from it. With a one sided tolerance, only one value counts anyway: Cmu for an upper limit, Cml for a lower one.
10 · Documentation
Calculate Cmk automatically and document it for audits
A single Cmk is quick to calculate. It gets harder with hundreds of tools and machines tested at fixed intervals. Then every value has to be traceable later: which raw data, which gauge, which tolerance, who tested and when?
The Manufacturing OS by CSP is a platform that brings process and quality data from manufacturing together in one system. The IPM module captures process data such as torque and angle directly at the station. QST manages tool tests and calculates the capability indices. Raw data, calculation and assessment stay linked and can be traced completely in an audit.
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FAQ
Frequently asked questions
How do you calculate the Cmk value?+
Calculate the mean x̄ and the standard deviation s from at least 50 measured values. Then compute Cmu = (USL minus x̄) / (3 times s) and Cml = (x̄ minus LSL) / (3 times s). The smaller of the two is the Cmk.
What is the difference between Cm and Cmk?+
Cm only relates the tolerance width to six standard deviations and tells you how tightly the machine scatters. Cmk also takes into account where the mean sits within the tolerance. If the machine is perfectly centred, both values are equal.
What is a good Cmk value?+
In practice, a machine is usually considered capable from a Cmk of 1.67. In general industry, 1.33 is sometimes accepted. What counts is always your customer requirement or your internal standard.
How many measured values do you need for Cmk?+
A machine capability study usually measures at least 50 parts that were produced without interruption and without readjusting the machine.
Can I calculate Cmk in Excel?+
Yes. Use AVERAGE and STDEV.S to get x̄ and s, then calculate Cmu, Cml and Cmk with simple formulas. Make sure to use STDEV.S, the sample standard deviation.
What is the difference between Cmk and Cpk?+
Cmk assesses the machine alone under constant conditions in a short term study. Cpk assesses the running process with all influences such as material, operators and tool changes. That is why the requirement for Cmk is higher.
Can Cmk be greater than Cm?+
No. Cmk can at most equal Cm. Both values are the same when the mean lies exactly in the middle of the tolerance.