Recently, I wrote about the process capability index and tolerance interval. Here, I’m writing about the relationship between the process capability index and sigma. The sigma number here relates to how many standard deviations the process window can hold.

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A +/– 3 sigma contains 99.73% of the normal probability density curve. This is also traditionally notated as the “process window.” The number of sigmas is also the *z-score*. When the process window is compared against the specification window, we can assess the process capability. When the process window is much narrower than the specification window and is fully contained within the specification window, we say that the process is highly capable. When the process window is larger than the specification window, we say that the process isn’t capable.

How much the process window is enclosed within the process specification window is explained by the process capability index. The most common process capability index is *Cpk* or *Ppk*. Here, we will consider *Ppk*.

*Ppk* is the minimum of two values:

Here µ is the mean, σ is the standard deviation, *LSL* is the lower specification limit, and *USL* is the upper specification limit. We’re splitting the process window into two and accounting for how centered the process is. If the process window isn’t centered compared to the process specification window, we penalize it by choosing the minimum of the two.

For convenience, let’s assume the equation below:

If we multiply both sides by 3, the equation becomes:

The value on the right side can be expressed as, “How many standard deviations are contained in the split process window?” This is also the sigma value or the *z-score*.

For example, if the *Ppk* is 1.00, then the z-score is 3.00. This means that the process window and the specification window overlap exactly. This corresponds to 99.73% of the curve. Please note that I’m assuming that the process is perfectly centered. Refer to this post for additional details on calculations for unilateral and bilateral capabilities.

In other words,

This relationship allows us to estimate the percent-conforming (percent under the curve) by just knowing the process capability index value. A keen reader may also notice the similarity to tolerance interval calculations. If we go back to the idea that sigma is the number of standard deviations that the split process window can accommodate, then we can replace sigma with *k1* and *k2* factors used for the tolerance interval calculations for unilateral and bilateral interval calculations.

A word of caution here is about the switcheroo that happened. The calculations we are doing are based on the normal probability distribution curve, and not the actual process probability distribution curve. The accuracy of our inferences will depend on how closely the actual process probability distribution curve matches the beautiful symmetric normal curve.

Always keep on learning.

*Published June 29, 2024, in Harish’s Notebook.*

## Comments

## Overly narrow treatment?

Given that capability confusion is somewhat common, does your narrow treatment of the topic not risk propagating this “confusion”?

Capability indexes, as I have no doubt you know, are reliable, or well-defined, indicators

if, and onlyif, the process data display a reasonable degree of consistency (i.e. stable or “in control” process). All these assumptions of probability models are misleading unless there is some reasonable evidence of process stability over time.Moreover, wouldn’t it also be advantageous to make a clear separation between Cp/Cpk as “capability indexes” and Pp/Ppk as “(past) performance indexes”?

Lastly, rather than getting bogged down in calculations how about keeping things simple with a primary use of time-series plots, control charts and histograms?

## Capability indices add no value

Capability indices are not a substitute for a control chart plus histogram. These statistics add no value, as my animation demonstrates:

https://www.linkedin.com/posts/dr-tony-burns-b040541_is-capability-as-much-nonsense-as-oee-cp-activity-7221256560197476354-3kOJ?

## Accuracy of Inferences

You said in your previous article (linked in this one)

The accuracy of our inferences will depend on how closely the actual process probability distribution curve matches the beautiful symmetric normal curve.

This sounds at odds in several ways with Dr. Deming's recommendations in Dr. Shewhart's second book**

"An inference, if it is to have scientific value, must constitute a prediction concerning future data. If the inference is to be made purely with the help of the distribution theories of statistics, the experiments that constitute the evidence for the inference must arise from a state of statistical control; until that state is reached, there is no universe, normal or otherwise, and the statistician’s calculations by themselves are an illusion if not a delusion. The fact is that when distribution theory is not applicable for lack of control, any inference, statistical or otherwise, is little better than conjecture. The state of statistical control is therefore the goal of all experimentation,"

**Shewhart, Walter A. (Walter Andrew), 1891-1967. Statistical Method from the Viewpoint of Quality Control. Washington: The Graduate School, The Department of Agriculture, 1939.

Thanks,

Allen

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