The regions labeled *CR* and *NS* represent misclassifications due to measurement error. *CR* represents conforming product that gets rejected, while *NS* represents nonconforming product that gets shipped. By integrating the bivariate normal distribution over the regions shown, we can estimate the *minimum* percentages for each type of misclassification. Figures 7 and 8 show these minimums as a function of the capability ratio and the *P/T* ratio.

Figure 7 shows the minimum percentage of nonconforming product that will get shipped for a given capability and *P/T* ratio when the process is operated up to its full potential. (The bottom two curves are for capabilities of 0.80 and 0.90.)

and *Bivariate Correlation = *

These ratios are related in the manner shown by the following equation. This means that knowledge of any two of these ratios will completely define the third.

The only way to avoid these misclassifications is to have a capability ratio large enough for the ellipse to fit within the square, or to have zero measurement error. So, if you depend upon inspection in order to ship conforming product, your target value for the *P/T *ratio has to be zero.

### Summary

So if you’re concerned with the value of your *P/T *ratio, you may not have progressed to the point of asking the right questions. The choice is yours. Are you going to continue to scrape the burnt toast, or are you going to learn how to stop burning the toast?

### Appendix: Finding the maximum process yields

Thus, the complement of the *ICC* compares the variance of the measurement process, *Var(E)*, with the variance of the stream of product measurements, *Var(X)*.

The proportion of nonconforming product (*N*) that gets rejected (*R*) by the specifications will be approximated by the volume under the bivariate model in the regions labeled *NR.*

Lean

## Using the Precision to Tolerance Ratio

### What does this ratio tell us?

As we learned last month, the precision to tolerance ratio is a trigonometric function multiplied by a scalar constant. This means that it should never be interpreted as a proportion or percentage. Yet the simple *P/T* ratio is being used, and misunderstood, all over the world. So how can we properly make use of the *P/T* ratio?

When the *ICC* falls below 0.20, the measurements contain very little information about the product. These fourth-class monitors are essentially useless for process improvement.

With appropriate changes in the limits of integration, we can use the equation above to find volumes for the regions *CR*, NS*,* and *NR* for any given combination of the* P/T* and capability ratios.

When we use probability models to characterize what to expect in practice, we should round off the results to parts per thousand. No probability model will ever describe reality out to parts per ten-thousand or beyond. While we may compute values out to more decimal places, and while the extra digits may be necessary for precise computations, these extra digits are meaningless when it comes to describing finite processes.

Inspection places a premium on having a good measurement system. When a process is operated unpredictably or off target, the misclassifications will increase beyond the minimums given above. The costs of improving the measurement system, plus the costs of inspection, rework, and scrap, will lower productivity without increasing the quality of the product shipped. I have had clients who were so comfortable with inspection that they put on more inspectors, increased the size of the rework department, and slowly went out of business.

Thanks,

Quality Digest