# Process Capability Analysis

**Process Capability Analysis**

Supposing that QC manager has corrected the process so that the mean is as desired, do you think that the refills will conform to their length specifications? The answer is NO.

In Fig, curve (a) represents the process before adjusting for the mean and (b) represents the process after adjusting the mean. Note that there is no change in the variability. Thus, we will continue to have rejections even after adjusting for mean unless the process variability itself is reduced.

At this stage, we have to answer two question: (1) What is the existing variability? (2) How much should we reduce it by? To answer these questions, we need the following definitions.

**Definition**. Total tolerance of a measurable quality characteristic, denoted by T, is given by the difference T = USL − LSL, where USL and LSL are the upper and lower specif ication limits, respectively.

For example, as LSL = 9.80 cms and USL = 10.20 cms for refill length problem, so, the total tolerance T = 0.4 cms in this case.

**Definition.** When a process is under statistical control, its process capability is given by 6σ, where is the process standard deviation.

The first question that we raised above can be answered by specifying an estimate of the process capability. Here, in our situation,

estimate of the process capability = 6R/d_{2}.

For refill length problem, the specification limits are 10 ± 0.2 cms. So, to avoid rejections, we must necessarily have a process for which the 3σ limits lie within the specification limits after setting the mean at the target (see Fig.).

In other words, the process capability must be less than the total tolerance. Therefore, the answer to the second question we raised above is that the process capability should not exceed the total tolerance.

**Definition.** The process capability ratio of a stable process is the ratio of total tolerance to the process capability and is denoted by C_{P}. That is,

Thus, by what we have said above, if Cp < 1, then the process is bound to produce rejections even when the mean is set on target. And, if Cp >=1, the rejection percentage will be almost zero, provided the mean is at the target.

An estimate of Cp can be obtained by substituting the estimate for in the above formula. For example, an estimate of Cp for the refill length problem under study is given by

Since the estimate of Cp is less than 1, we may infer that the ref ill length process is not capable. More generally, even if we get the estimate of Cp slightly more than 1, we may still consider the process incapable. This is because our estimate for Cp may be an underestimate due to sampling fluctuations.