6.1.6. What is Process Capability?

6. Process or Product Monitoring and Control
6.1. Introduction

6.1.6. What is Process Capability?

A process capability index uses both the process variability and the process specifications to determine whether the process is "capable"

Process capability compares the output of an in-control process to the specification limits by using capability indices.

Process Capability Indices

We are often required to compare the output of a stable process with the process specifications and make a statement about how well the process meets specification. To do this we compare the natural variability of a stable process with the process specification limits.

A capable process is one where almost all the measurements fall inside the specification limits. This can be represented pictorially by the plot below:

There are several statistics that can be used to measure the capability of a process: C_p, C_pk, C_pm.

Most capability indices estimates are valid only if the sample size used is ‘large enough’. Large enough is generally thought to be about 50 independent data values.

The C_p, C_pk, and C_pm statistics assume that the distribution of the data is normally distributed. Assuming a two sided specification, if m and s are the mean and standard deviation, respectively, of the normal data and USL, LSL are the upper and lower specification limits, then the population capability indices are defined as follows:

Definitions of various process capability indices

Sample estimates of capability indices

Sample estimates for these indices are given below. (Estimates are indicated with a "hat" over them).

The estimator for the C_pk can also be expressed by C_pk = C_p(1-k) where k is a scaled distance between the midpoint of the specification range, m, and the process mean, m.

To get an idea of the value of the C_p for varying process widths, consider the following plot

This can be expressed numerically by the table below:

Translating capability into "rejects"

where ppm = parts per million and ppb = parts per billion.

Thus far we discussed the situation with two spec. limits, the USL and LSL. This is known as the bilateral or two-sided case. There are many cases where only the lower or upper specifications are used. Using one spec limit is called unilateral or one-sided. The corresponding capability indices are

One-sided specifications and the corresponding capability indices

where m and s are the process mean and standard deviation, respectively.

Estimates of C_pu and C_pl are obtained by replacing m and s by xbar and s, respectively. The following relationship holds

C_p = (C_pu + C_pl) /2.

This can be represented pictorially by

Note that we also can write:

C_pk = min {C_pl, C_pu}.

Confidence intervals for indices

Confidence Limits For Capability Indices

Assuming normally distributed process data, the sample C_p follows a Chi-square distribution and C_pu and C_plhave distributions related to the non-central t distribution. Fortunately, approximate confidence limits related to the normal distribution have been derived. The distribution of the C_pk index is based on a bivariate t distribution but the confidence limits are much easier obtained by assuming an approximately normal distribution.

The resulting formulas for confidence limits are given below:

Zhang (1990) derived the exact variance for the estimator of the
C_pk as well as an approximation for large n. The reference paper is: Zhang, Stenback and Wardrop: Interval Estimation of the process capability index, Communications in Statistics: Theory and Methods, 19(21), 1990, 4455-4470.

The variance is obtained as follows:

His approximation is given by:

It is important to note that the sample size should be at least 25 before these approximations are valid. Another point to observe is that variations are not negligible due to the randomness of capability indices.

An example

Capability Index Example

What you can do with non-normal data

What happens if the process is not normally distributed?

The indices that we considered thus far are based on normality of the process distribution. This poses a problem when the process distribution is not normal. Without going in the specifics we can list some remedies.

     1. Transform the data so that they become normal. A
          popular transformation is the Box-Cox transformation
     2. Use or develop another set of indices, that apply to
          nonnormal distributions. One statistic is called C_npk (for
          non-parametric C_pk). Its estimator is calculated by

          where p(0.995) is the 99.5th percentile of the data
          and p(.005) is the 0.5th percentile of the data.
     3. There are two flexible families of distribution that are
          often used: the Pearson and the Johnson families.
          The sample skewness and kurtosis are used to pick a
           model and process variability is then estimated.
      4. Use mixture distributions to fit a model to the data.
           A mixed Poisson model has been used to develop the
           equivalent of the 4 normal based capability indices.

There is, of course, much more that can be said about the case of the nonnormal data. However, if a Box-Cox transformation can be successfully performed, one should be encouraged to use it.