What are Capability Indices
Terminology and Actual Versus Estimated Sigma
Before describing capability indices in detail, several terms must be defined. USL stands for Upper Specification Limit and LSL for Lower Specification Limit. Midpoint is the center of the specification limits. Midpoint is often referred to as the nominal value or target. Tolerance is the distance between the specification limits (TOLERANCE = USL – LSL).
The standard deviation for the distribution of individuals, one important variable in all the capability index calculations, can be determined by either of two ways. It may be calculated or estimated. For an explanation of the concept of standard deviation, see Chapter 1.
Standard deviation may be calculated using the standard statistical formula:
σ (Sigma) symbolizes population standard deviation, N is the population size, x is an individual measurement, and µ is the mean of the population. Population means all parts being produced, not just a sample.
Usually, we work with a sample of the population, since this is more practical. In this case, the formula for standard deviation is:
s symbolizes sample standard deviation, and n is the sample size. When more than 30 samples are taken, the above formulas for population and sample standard deviation yield the same result. Therefore, we will use the term Sigma as we continue our discussion.
Standard deviation may also be estimated using R (the average of the subgroup ranges) and a constant that has been developed for this purpose. The formula for estimating Sigma is:
σˆ (Sigma hat or estimated Sigma) symbolizes estimated standard deviation. R is the average of the subgroup ranges for a sample period when the process was in control. The constant d2 varies by subgroup size and is listed in the Appendix, Table A-1.
It is important to remember that the process must be normally distributed and in control to use the estimated value of Sigma. If both of these conditions are not met, the estimated value is not valid. If the process is normally distributed and in control, either method is acceptable and usually yields about the same result. Also, remember that neither actual nor estimated Sigma used in calculating capability will be meaningful if the process is inherently abnormal. See 5.6 Capability Measures for Abnormal Distributions for information on measuring the capability of abnormal processes.
If you have both estimated and actual capability indices available, choose one method and stay with it. Avoid the temptation to look at both and choose the one that is better, since this will introduce variation in results.
The most commonly used capability indices are CP and CpK. CP, which stands for capability of process, is the ratio of tolerance to 6 Sigma. The formula is:
he 6σ in the CP formula comes from the fact that, in a normal distribution, 99.73% of the parts will be within a 6σ (±3σ) spread when only random variation is occurring.
The CP for the sample distribution in Figure 5.4.1 is:
As you can see from the CP formula, values for CP can range from near 0 to very large positive numbers. When CP is less than 1, tolerance is less than the 6σ spread of the distribution. When CP is greater than 1, tolerance is greater than the 6σ spread of the distribution. The greater the number, the better the CP index.
CP is only a measure of the dispersion or spread of the distribution. It is not a measure of centeredness (where the distribution is in relation to the midpoint). Figure 5.4.2 shows two distributions for the same specification limits. Both distributions have a CP of 1.25, but one shows almost all parts being within specification, and the other shows a significant number of parts out of specification. This is why CP is never used alone as a measure of capability. By itself, it cannot indicate whether the process is capable. CP only shows how good the process could be if centered. Therefore, CP is usually used with CpK.
While CP is only a measure of dispersion, CpK is a measure of both dispersion and centeredness. That is, the formula for CpK takes into account both the spread of the distribution and where that distribution is in regard to the specification midpoint. The formula is:
CpK = The lesser of:
Because we choose the lesser of the two values calculated, we find out how capable our process is on the worst side (the tail closest to the specification limit).
Using the example used in the CP calculation,
The greater the CpK value, the better. A CpK greater than 1.0 means that the 6σ (±3σ) spread of the data falls completely within the specification limits. A CpK of 1.0 means that one end of the 6µ spread falls on a specification limit. A CpK between 0 and 1 means that part of the 6σ spread falls outside the specification limits. A negative CpK indicates that the mean of the data is not between the specification limits. To help you visualize what CpK indicates, Figure 5.4.3 shows the distributions for 4 different CpK values. A small CpK value is due to a process being off center from the specification midpoint, spread out, or both. The CpK index will not indicate whether the process is centered above or below target; to see this, look at the mean of the samples compared to the midpoint (target or nominal). Or look at the histogram relative to the specification limits.
Fig. 5.4.3 The greater the CpK value, the better. CpK is a measure of both spread and centeredness. It looks at the 3 sigma limit of the tail of the curve farthest from the specification midpoint, and indicates how far this is from the specification limit.
Since a CpK of 1.0 indicates that 99.73% of the parts produced are within specification limits, in this process it is likely that only about 3 out of 1,000 need to be scrapped or rejected. Why bother to improve the process beyond this point, since it will produce no reduction in scrap or reject costs? Improvement beyond just meeting specification may greatly improve product performance, cut warranty costs, or avoid assembly problems.
Many companies are demanding CpK indexes of 1.33 or 2.0 of their suppliers’ products. A CpK of 1.33 means that the difference between the mean and specification limit is 4σ (since 1.33 is 4/3). With a CpK of 1.33, 99.994% of the product is within specification. Similarly a CpK of 2.0 is 6σ between the mean and specification limit (since 2.0 is 6/3). This improvement from 1.33 to 2.0 or better is sometimes justified to produce more product near the optimal target. Depending on the process or part, this may improve product performance, product life, customer satisfaction, or reduce warranty costs or assembly problems.
Continually higher CpK indexes for every part or process is not the goal, since that is almost never economically justifiable. A cost/benefit analysis that includes customer satisfaction and other true costs of quality is recommended to determine which processes should be improved and how much improvement is economically attractive.
Relationship between CP and CpK
By looking at the formulas, you can see the relationship between CP and CpK. CpK cannot be greater than CP for a process. Only when the mean is exactly centered on the specification midpoint is CpK = CP. Thus, CP is valuable as an indicator of how much better the CpK could be if the process were set up so that the center of the distribution were closer to the specification midpoint. CP answers whether the distribution of individuals could fit within the tolerances (if centered). CpK answers whether the distribution does.
CR is a less-frequently-used substitute for CP. It is simply the inverse of CP. Thus, the formula is:
For our example:
Since CR is the inverse of CP, a CP of 1.33 equals a CR of 0.75. The smaller the CR value, the better. CP is more frequently used than CR since it is much easier to compare CP to CpK.
Another indication of capability (actually an extension of CpK) is Zmax/3. CpK looks at the tail of the distribution that is closer to the specification. Zmax/3 is a CpK for the tail of the distribution that is further from the specification. Another way of expressing this is that CpK is Zmin/3 (see the formula for CpK).