The spread of mass production in the previous century has necessitated the implementation of statistical methods in order to verify and control manufacturing processes. The statistical measurement of processes provides key inputs for engineers and decision makers, that’s why Statistical Process Control (SPC) has been elaborated during the previous decades.

In the quality management literature, the measurement and evaluation of machines and processes are called capability studies.

Measuring the capability of machines and processes have several benefits:

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.

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.

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:

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 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: C_m and C_mk. Both represent the capability of a machine, but in different ways.

Machine capability (C_m) is the relation between the spread of the machine and the tolerance range (e.g. C_m = Tolerance / Spread). By other words, C_m 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 C_m 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:

C_m = (USL – LSL) / 6 * Ϭ, or T / 6 * Ϭ

Important: the higher your machine variation is (sigma), the lower your C_m will be. The higher the C_m is, the more stable your process will be (from machine side).

The machine capability index (C_mk) not only represents the correlation between the spread and the tolerance range, but also takes the location of the distribution into account.

C_mk = min [ (USL - μ) / 3 * Ϭ ; (μ - LSL) / 3 * Ϭ ]

Important: the higher your machine variation is (sigma), the lower your C_mk will be. On top of that, the higher the offset is, the lower your C_mk will be. Offset means the distance between the middle of the tolerance range and the process mean.

The process capability study is a long-term evaluation of a process that takes all influencing factors into account, for example:

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 (C_p, C_pk), 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 or 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 C_m is similar to C_p (so as with C_mk and C_pk) as the mathematical background is the same, the differences are discussed in the 'Capability Studies' table in the beginning of the section.

Process capability (C_p) is the relation between the spread of the process and the tolerance range (e.g. C_p = Tolerance / Spread). By other words, C_p 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 C_m 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:

C_p = (USL – LSL) / 6 * Ϭ, or T / 6 * Ϭ

Important: the higher your process variation is (sigma), the lower your C_p will be. The higher the C_p is, the more stable your process will be.

Process capability index (C_pk) not only represents the correlation between the spread and the tolerance range, but also takes the location of the distribution into account.

C_pk = min [ (USL - μ) / 3 * Ϭ ; (μ - LSL) / 3 * Ϭ ]

Important: the higher your machine variation is (sigma), the lower your C_pk will be. On top of that, the higher the offset is, the lower your C_pk will be. Offset means the distance between the middle of the tolerance range and the process mean.

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.

Having a low C_pk in our manufacturing affect scrap rate and defect cost. Even with a 4 sigma capable process (which is equal to C_pk = 1.33), the measured characteristic will be out of the specification (tolerance range) on 63 ppm of the products.

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 6 sigma 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).

While C_p and C_pk indices are calculated within sub-groups (e.g. consecutive measurements in one shift), P_p and P_pk 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.

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 C_m, C_mk, C_p and C_pk 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 manufacturing environment: