Using the Precision to Tolerance Ratio

What does this ratio tell us?

Published: Tuesday, September 5, 2023 - 11:03

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?

The usual motivation behind the computation of the P/T ratio is a desire to determine if a measurement procedure is adequate for a given production process. And there are two ways that measurements support a production process: They can be used to scrape the burnt toast (inspection), or they can be used to learn how to stop burning the toast (process improvement). If the measurement system is adequate to allow us to improve the process, then we can often get to the point where we no longer have to depend upon inspection to ship conforming product. So, of these two ways that measurements support production, the role of process improvement is the more critical in the long term.

Will the measurements support process improvement?

To assess the adequacy of the measurement process for the role of process improvement, we use the maximum process yield. This is that fraction of the product produced that is both conforming and gets shipped when the production process is operated predictably and on-target.

When a production process is not operated on-target, or when it is not operated predictably, it is not being operated up to its full potential. And when a process is not operated up to its full potential, then its actual yield will be less than the maximum process yield. Yet here the maximum process yield may still be used as a conservative estimate of the hypothetical process potential; it will give us an idea of what the process yield could be when operated up to its full potential.

The maximum process yield depends upon three quantities: the specified tolerance; the variation in the measurement process, SD(E); and the variation in the stream of product measurements, SD(X).

Figure 1: The maximum process yield depends upon three quantities

Three ratios are commonly used to compare the quantities on the corners of the triangle. The capability ratio, Cp, compares the specified tolerance with the (within subgroup) standard deviation of the product measurements:

The precision to tolerance ratio compares the standard deviation of the measurement process with the specified tolerance.

And the intraclass correlation coefficient (ICC) defines that proportion of the variation in the stream of product measurements, Var(X), that can be directly attributed to the variation in the product steam, Var(Y):

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).

So that the ratio of SD(E) to SD(X) will be given by:

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.

So, while none of these three ratios will tell the whole story by itself, any two of them may be used to determine the maximum process yield. Figure 2 shows how the P/T ratio and the capability ratio combine to determine the maximum process yield. The horizontal axis shows values of the P/T ratio ranging from 0.00 to 1.00, while the vertical axis shows capability ratios ranging from 0.50 to 1.20. The values in the graph are the maximum process yields, in parts per 1,000. These are the number of conforming units shipped per 1,000 units produced.

Figure 2: How maximum process yields change with P/T and capability

For example, if your P/T ratio is 0.60 and your capability ratio is 0.80, your process will have a maximum process yield of about 980 ppt. Operating predictably and on-target will result in about 980 conforming units shipped per 1,000 units produced. Operating at less than full potential will result in a lower yield.

Figure 3 shows the information of figure 2 in a different format and extends the coverage to lower capability ratios. Here, each curve represents a fixed value for the capability ratio while the maximum process yields are shown on the vertical axis.

Figure 3: Capabilities effectively determine maximum yields

The flatness of the curves in figures 2 and 3 shows that the P/T ratio has very little impact upon maximum process yields. The yields are almost completely determined by the capability ratios. So, we don’t have to have a small P/T ratio in order to have a high maximum process yield. We can have very high yields even with a large P/T ratio. This means that knowing the P/T ratio doesn’t tell us anything useful about the maximum process yield.

The usefulness of the measurement system

For the role of process improvement, we need a measurement system that allows us to detect those changes in our process that are large enough to be of economic impact. To see how a measurement system affects the ability of a process behavior chart to detect process changes, we will use the intraclass correlation coefficient.

Using published tables of the power functions for process behavior charts, we can compute the probabilities of detecting various sized shifts for various values of the intraclass correlation. Here, we shall consider detecting a three-standard-error shift within 10 subgroups of when that shift occurs. Figure 4 shows two curves: The lower curve is for the use of detection rule one alone, while the upper is for the use of all four Western Electric zone tests. The interesting thing about both curves is how slowly they drop off as the intraclass correlation goes to zero.

Figure 4: The ability to detect economic changes

When the intraclass correlation is greater than 0.80, the measurement system is a first-class monitor for that production process. A process behavior chart using detection Rule 1 alone has better than a 99% chance of detecting a three-standard error shift.

When the intraclass correlation is between 0.80 and 0.50, the measurement system is a second-class monitor for that production process. Here, a process behavior chart using detection Rule 1 alone will have better than an 88% chance of detecting a three-standard error shift. Using all four detection rules together gives a 100% chance of detecting a three-standard error shift within 10 subgroups.

When the intraclass correlation is between 0.50 and 0.20, the measurement system is a third-class monitor for that production process. Here, a process behavior chart using all four detection rules still has better than a 91% chance of detecting a three-standard error shift in the production process within 10 subgroups.

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.

Figure 5 is an extension of figure 2 showing the curves that separate these four classes of monitors. While P/T ratios below 0.30 will correspond to first-class monitors, larger P/T ratios may correspond to various classes of monitors. Since first-, second-, and third-class monitors are all useful for process improvement, the P/T ratio does not describe the relative utility of a measurement process.

Figure 5: P/T does not determine ICC

So, the P/T ratio does not limit the maximum process yields. Neither does it characterize the relative usefulness of the measurements for keeping process behavior charts. But does it tell us anything about using the measurements to scrape the burnt toast?

Inspection

When sorting nonconforming product from conforming product we will want perfect measurements. Any amount of measurement error will create the possibility of misclassifications where some conforming product gets rejected and some nonconforming product gets shipped. To illustrate how this happens, we return to the model for our measurements.

The product measurements, X, will consist of the product values, Y, plus measurement errors, E. The relationship between X and Y may be modeled with a bivariate normal distribution. The ellipse in figure 6 shows the three-sigma contour of a bivariate normal distribution having an ICC value of 0.80 and a capability ratio of 0.60.

Figure 6: Relationship between X and Y when ICC = 0.80

The region labeled CS represents the maximum process yield. As the capability ratio increases, the specifications get wider relative to the ellipse and more of the ellipse will fall within the specification square. As Var(E) gets smaller, the ellipse gets narrower and the minor axis shrinks.

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: Minimum percentages of nonconforming product shipped

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.)

Figure 8: Minimum percentages of conforming product rejected

Figure 8 shows the minimum percentage of conforming product that will get rejected for a given capability and P/T ratio when the process is operated up to its full potential.

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

The only way to maximize your process yield and minimize the costs of misclassifications is to operate your process predictably and on target. This requires that you use your measurements for process improvement. Fortunately, just about any old measurement process will serve to do this. Even second- and third-class monitors will work with process behavior charts. As you use these charts, you’ll discover assignable causes of exceptional variation. As these assignable causes are controlled, the process capability will increase, the process yields will increase, and the misclassifications will dwindle. This will reduce inspection, scrap, and rework costs, which will increase your productivity while improving product quality. I have clients who, when operating in this manner, have turned companies around, and others who have taken over markets by producing the best quality at the lowest cost.

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.

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

The following is provided for those interested in duplicating the results shown in the graphs. The relationship between the product values, Y, and the product measurements, X, can be modeled using a bivariate normal distribution as shown in figure 6. In this model, the minor axis of the ellipse is a function of the measurement error.

The proportion of conforming product (C) that gets shipped (S) will be approximated by the volume under the bivariate model in the region labeled CS. This value will be the maximum process yield.

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.

The proportion of conforming product (C) that gets rejected (R) by the specifications will be approximated by the volume under the bivariate model in the regions labeled CR. This is a misclassification that occurs due to measurement error.

And the proportion of nonconforming product (N) that gets shipped (S) will be approximated by the volume under the bivariate model in the regions labeled NS. This is the other misclassification that occurs due to measurement error.

 Without loss of generality we may assume:

Mean(X) = Mean(Y) + Mean(E) = 0

Var(X) = Var(Y) + Var(E) = 1.0 + Var(E)

and Bivariate Correlation =

Then, for a given P/T ratio and a given capability ratio, we may find the ICC value and compute the maximum process yield by integrating the resulting bivariate normal distribution with respect to both X and Y over the region labeled CS (i.e., between the specification limits).

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 1,000. No probability model will ever describe reality out to parts per 10,000 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.

About The Author

Donald J. Wheeler

Dr. Wheeler is a fellow of both the American Statistical Association and the American Society for Quality who has taught more than 1,000 seminars in 17 countries on six continents. He welcomes your questions; you can contact him at djwheeler@spcpress.com.