We distinguish between machine and process capability studies. Both rest on the same methodological basis,
however the intentions are different. On top of that, we also need to differentiate short-term and long-term studies.
In order to understand capability, we have to interpret some basic terms.
Capability study: an analysis procedure, which is performed to measure and analyse the capability of a
process or machine. During the analysis, the relevant process parameter, or the characteristic of the products, that is affected by the
given process is measured. After having a data set, we can perform various calculations to evaluate the results compared to a specification.
||Short-term (e.g. one continuous production run)
||Long-term (e.g. continuous measurement of process capability in serial production)
||To reveal the machine-specific effects on the production
||To discover all process-affecting factors and coefficients, that have impact on the capability of the manufacturing process
||New machine setup and optimization
||New process setup, process optimization, Statistical Process Control (SPC)
|What does an inaccurate number mean?
||Your machine itself is generating unacceptable variation and it’s incapable to produce in range.
||Your process is incapable due to an affecting factor (machine, material, environment, etc.)
|Number of values taken
||min. 50 units (100 units preferred)
min. 25 units if Cmk ≥ 2.0
|min. 125 units
(hundreds or thousands of units preferred)
||Cmk ≥ 1.67
||Cpk ≥ 1.33
||Cm and Cmk
||Cp and Cpk and Pp and Ppk
Stability of process: in statistical process control, stability means low variation. A stable process is
precise with low variance, and the measured characteristics are in an acceptable range.
Capability of process: the process is capable, if all requirements (based on specification) are met.
Upper Specification Limit (USL): the highest permissible value of a given characteristic.
Lower Specification Limit (LSL): the lowest permissible value of a given characteristic.
Tolerance range (T): the distance between USL and LSL (T = USL – LSL).
Spread: the quantity (amount) of variability in a given population.
Mean / Average (μ): the arithmetic mean of the data set.
Median: the middle value of the data set.
Variance (Ϭ2): is calculated by taking the differences between each value of the data set and the mean,
squaring the differences and dividing the sum of the squares by the number of values in the data set.
Standard deviation (Ϭ or sigma): is equal to the square root of variance.
Important: as we calculate the mean of random samples, instead of calculating the mean of the whole
population, we have to use n-1 (degrees of freedom) in the denominator during the calculation of variance and standard deviation.
Calculation of variance and standard deviation (Source: qMindset.com)
We have to note, that stability or capability by itself is not enough. The process needs to have both to
produce proper characteristics consistently. Features of a stable and capable process:
- Mean is at the nominal specification, with very low variance.
- No typical trend is visible (no systematic variation in the means).
- Sample variation and the variation of the total population are not different significantly.
Let’s take some examples (see chart):
Drunken bowman: he shots on the target randomly, the variation of his shots is high, and on top of that,
the location of the shots is incorrect, as he barely hits the board. He is neither capable, nor stable.
Lucky drunken bowman: although the variation of his shots is still high, somehow he hits the board every
time. He is capable, but not stable.
Skilled bowman with wrong glasses: his skill is superb, his hits are very near to each other, but because
of the distorting lens in his glasses, the shots systematically go to the upper right side of the board. He is stable, but not capable.
Robin Hood: with his perfect skill, he is able to systematically find the middle of the target. His shots
have very low variance, and the mean is exactly in the middle. He is both stable and capable.
The visualization of stability and capability (Source: qMindset.com)
Machine capability study:
The machine capability study is an evaluation that represents only the internal production capabilities and
characteristics of the machine (cycle time, tooling, voltage, current, etc.) without deeply considering the external influencing factors (warm-up,
environmental temperature, personnel, etc.). The intention of this short-term study is to analyze the influence of the machine-specific impacts
on the manufacturing process.
For the calculation of machine capability, we use two major indices: Cm and Cmk. Both represent the
capability of a machine, but in different ways.
Machine capability (Cm) is the relation between the spread of the machine and the tolerance range (e.g.
Cm = Tolerance / Spread). By other words, Cm indicates the number of times the width of the spread fits into the complete tolerance range.
It does not take into account where our distribution of data is located to the mean or upper- and lower specification limits, so even if
some of your values are near to USL or LSL, the Cm number remains the same.
Statistical software are calculating the spread (and pull a distribution curve over the spread) by
using the quantiles of the distribution, however a manual method is much easier and still usable. Instead of using quantiles, we simply
use the standard deviation (sigma) of the data set. By this method our formulae is the following:
Cm = (USL – LSL) / 6 * Ϭ, or T / 6 * Ϭ
Important: the higher your machine variation is (sigma), the lower your Cm will be. The higher the Cm is,
the more stable your process will be (from machine side).
Calculation of Cm (Source: qMindset.com)
The machine capability index (Cmk) not only represents the correlation between the spread and the tolerance
range, but also takes the location of the distribution into account.
Cmk = min [ (USL - μ) / 3 * Ϭ ; (μ - LSL) / 3 * Ϭ ]
Important: the higher your machine variation is (sigma), the lower your Cmk will be. On top of that, the
higher the offset is, the lower your Cmk will be. Offset means the distance between the middle of the tolerance range and the process mean.
Calculation of Cmk (Source: qMindset.com)
Process capability study:
The process capability study is a long-term evaluation of a process that takes all influencing factors
into account, for example:
- Machine (cycle time, adjusted parameters, warm-up time, etc.)
- Material (raw material, sub-assembly, etc.)
- Man (different operators in different shifts, different handling, etc.)
- Method (various pre-processing of part before the studied process, etc.)
- Environment (humidity, temperature, etc.).
All of these influencing factors affect the variation and stability of the process. Combining this
information with a larger sample size (long-term study), the process capability study provides a comprehensive evaluation about what we
can expect from our analyzed process.
Before calculating the process capability indices (Cp
), we should perform a study of process stability, to check the variances (sigma, means,
distribution, etc.) in sub-groups, and between sub-groups by doing F-test
ANOVA (analysis of variances)
. This pre-analysis already gives us hints about the stability of the process, and we can see if there is an
external factor, which takes a higher impact on our production process. Note: the calculation of Cm
is similar to Cp
(so as with Cmk
) as the mathematical
background is the same, the differences are discussed in the 'Capability Studies' table in the beginning of the section.
Process capability (Cp) is the relation between the spread of the process and the tolerance range (e.g. Cp
= Tolerance / Spread). By other words, Cp indicates the number of times the width of the spread fits into the complete tolerance range. It does
not take into account where our distribution of data is located to the mean or upper- and lower specification limits, so even if some of your
values are near to USL or LSL, the Cm number remains the same.
Statistical software are calculating the spread by using the quantiles of the distribution, however a
manual method is much easier and still usable. Instead of using quantiles, we simply use the standard deviation (sigma) of the data set. By
this method our formulae is the following:
Cp = (USL – LSL) / 6 * Ϭ, or T / 6 * Ϭ
Important: the higher your process variation is (sigma), the lower your Cp will be. The higher the Cp is,
the more stable your process will be.
Calculation of Cp (Source: qMindset.com)
Process capability index (Cpk) not only represents the correlation between the spread and the tolerance
range, but also takes the location of the distribution into account.
Cpk = min [ (USL - μ) / 3 * Ϭ ; (μ - LSL) / 3 * Ϭ ]
Important: the higher your machine variation is (sigma), the lower your Cpk will be. On top of that, the
higher the offset is, the lower your Cpk will be. Offset means the distance between the middle of the tolerance range and the process mean.
Calculation of Cpk (Source: qMindset.com)
By using a statistical software, it is very easy to calculate the Cp and Cpk values of a population (traditional spreadsheets are also usable). On top
of that, such a software supports to visualize the results and the histogram, so it helps to see through the process capability.
Example of Cp and Cpk calculation (Source: qMindset.com)
Having a low Cpk in our manufacturing affect scrap rate and defect cost. Even with a 4 sigma capable
process (which is equal to Cpk = 1.33), the measured characteristic will be out of the specification (tolerance range) on 63 ppm of the
To keep a consistently reliable manufacturing chain, the continuous abatement of process capability
must be considered as well. What does it mean?
The variance of every process increases over time (as a Murphy law) that must be kept in bay. If a machine and a process is designed to be capable for a
production, it does not mean, that the system will be able to keep this 6 sigma level on the long-term. Based on statistics, this shift
of process capability is estimated to 1.0 - 1.5 sigma, so a process with 6 sigma will be resulting defects on a 4.5 – 5.0 sigma level over years (see short-term and long-term adjusted sigma table).
The correlation of various sigma levels, Cpk, and ppm (Source: qMindset.com)
Some notes about Process performance indices (Pp, Ppk):
While Cp and Cpk indices are calculated within sub-groups (e.g. consecutive measurements in one shift), Pp
and Ppk are calculated by considering an unstable process with a systematic variation (e.g. shift-to-shift or batch-to-batch process variation).
Process performance indices calculate with the overall standard deviation (sigma) of the whole data set.
Visualization of process performance (Source: qMindset.com)
It is very important to note, that the capability of the measuring device critically affect both machine
and process capability. Before doing a machine- or process capability study, always perform an MSA or GRR% evaluation. If the chosen device is
not capable to measure the given characteristics (due to low GRR% or low "ndc" number), it mustn’t be used, as it misleads us, resulting
unreliable Cm, Cmk, Cp and Cpk values. On top of that, your sample size has a major impact on the confidence level of the study: the higher
the sample size, the more it represents the whole manufacturing population (leads to more reliable process capability study).
It is crucial to determine the right technology and the right machine for the given process. If this is
not done right during the APQP framework, the machine capability will be low even after all possible improvements, resulting high scrap rates.
To avoid this upfront, the DMADV SixSigma methodology (Define, Measure, Analyze, Design, and Verify) is highly advised.
It may happen, that the given tolerance ranges are set much stricter, than really needed. To change the
specification, an engineering change procedure (e.g. engineering change request) must be started, with the approval of internal engineering,
and the release of the customer.
Several statistical software help us by choosing the right distribution model that fits to our process.
Otherwise it would be very difficult, and would take a long time to calculate manually. Some examples for typical distributions in a
Typical distributions in a manufacturing environment (Source: qMindset.com)