In a class last month I was asked to explain a number that occurs in some measurement system evaluations and which is known as the precision to tolerance ratio (P/T ratio). As I will show in this column, it turns out to be related to the capability ratio.
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We will need some notation in what follows. So, let the product measurements be denoted by X. These product measurements may be thought of as consisting of two components. The value of the item being measured may be denoted by Y, while the error of the measurement may be denoted by E. Thus, X = Y + E. If we think about these quantities as variables, then the variation in the product measurements, Var(X), can be thought of as:
where Var(Y) is the variation in the stream of product values, and Var(E) is the variation in the stream of measurement errors.
The central purpose of a measurement system evaluation is to obtain an estimate of Var(E). Of course, once we have such an estimate the question of how to use it will arise. For more than 40 years it has been common to compare the standard deviation of measurement error with the specified tolerance for a particular product in an attempt to determine the relative utility of the measurement system for that particular product. If we denote the standard deviation of the measurement system by SD(E), and denote the specified tolerance by [ USL – LSL ], then the P/T ratio is commonly computed as:
Prior to the 1990s, the number 5.15 was used rather than the number 6.00. In either case, gauge R&R studies commonly define P/T values that are less than 0.10 to be good, while P/T ratios in excess of 0.30 are said to be inadequate. To make sense of the P/T ratio, we will need to establish how it is related to the interclass correlation coefficient and the capability ratio.
In my December 2010 column, I showed how the traditional and theoretically correct measure of relative utility is the intraclass correlation coefficient:
This ratio defines that proportion of the variation in the product measurements that can be attributed to the product stream, and it is also the complement of that proportion of the variation in the product measurements that can be attributed to the measurement system. From the last expression above we can establish that:
Taking the square root of each side, we get:
Next, we need to recall that the capability ratio is defined as the specified tolerance divided by six times the standard deviation of the product measurements, X.
Using the relationship between SD(X) and SD(E) given above, we can rewrite the capability ratio in terms of measurement error and the intraclass correlation coefficient as:
For a given measurement system and a given set of product specifications, the value for Lambda will remain fixed, and both the capability ratio and the interclass correlation coefficient will depend upon Var(Y). Specifically, as improvements are made to the production process, Var(Y) will decrease, the capability ratio will increase, and the intraclass correlation coefficient will get smaller. In the limit, as ICC goes to zero, the capability ratio will increase toward an upper bound defined by the value for Lambda. However, since Var(Y) cannot go to zero, neither can ICC, and Lambda remains a forever unreachable upper bound for the capability ratio.
Inspection of the expression above will show that Lambda is simply the inverse of the P/T ratio. This means that the P/T ratio is the inverse of the unobtainable upper bound for the capability ratio. It is the inverse of an impossible value. The following expression summarizes the relationship between the capability ratio, the P/T ratio, and the intraclass correlation coefficient:
Now that we know how the P/T ratio is related to other quantities, we turn to the question of whether we can use it to define when a measurement system is adequate. The general guideline found in various gauge repeatability and reproducibility (R&R) studies is that good measurement systems will have a P/T ratio that is less than 0.10, while merely adequate measurement systems will have a P/T ratio that is less than 0.30.
Asking for a P/T ratio to be smaller than 0.30 is like asking for an upper bound of 3.33 or greater for your capability ratios. While having the ability to compute such capability ratios is good, we can, and often do, work with measurement systems that cannot deliver such rarefied capability ratios. Even if it made sense to define the relative utility of a measurement system in terms of an upper bound on the capability ratios, there is hardly anything special about a capability ratio of 3.33 except that it is larger than what we typically encounter.
The arbitrary nature of the guideline above may be seen in the following manner. Those gauge R&R studies that use the above guideline for the P/T ratio also tend to include a second guideline that requires a quantity known as the "combined R&R ratio" (GRR) to be less than 0.30 for an adequate measurement system. This combined R&R ratio is defined as:
Inspection of the earlier expression for the interclass correlation coefficient will show that:
So that when the GRR ratio has a value of 0.30 the ICC value will be 0.91. Using the guidelines that define an adequate measurement system as one that has a P/T ratio of 0.30 and also has a GRR ratio of 0.30, we can use the formula given earlier to compute a value for the corresponding capability ratio. Recall that:
Here we see that when P/T = 0.30 and GRR = 0.30, then the capability ratio will be 1.00. Once you get to this point you are in a bind. Any reduction in Var(Y) that would increase the capability ratio (a desirable outcome) will simultaneously increase the GRR value into the range where the guidelines will characterize the measurement system as inadequate (an undesirable outcome). This use of guidelines for multiple characteristics effectively over-constrains the definition of an acceptable measurement system.
Hence, the use of the P/T ratio to characterize the relative utility of a measurement system for a particular product is both inappropriate and ineffective.
The correct, sound, and appropriate measure of relative utility is the interclass correlation coefficient. In my December 2010 column I showed how measurement systems having intraclass correlations in excess of 0.20 (GRR values as large as 0.89) are able to track process changes. This effectively shows that the guidelines commonly used in gauge R&R studies are excessively conservative and arbitrary. Learn how to use the right measure, and you will no longer have to depend upon arbitrary guidelines to interpret the inverse of impossible values.
Dr. Donald J. Wheeler is a Fellow of both the American Statistical Association and the American Society for Quality, and is the recipient of the 2010 Deming Medal. As the author of 25 books and hundreds of articles, he is one of the leading authorities on statistical process control and applied data analysis. Contact him at www.spcpress.com.
