Honest Precision to Tolerance Ratios |

Quality Digest71 day(s) ago

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Honest Precision to Tolerance Ratios |

Quality Digest71 day(s) ago

The precision to tolerance ratio is commonly used to characterize the usefulness of a measurement system While this ratio is appealingly simple it overstates the damage due to measurement error In this paper we show how to compute honest precision to tolerance ratios that correctly describe each of three different guard-banded sets of manufacturing specifications By presenting these options rather than using single ratio and artificial guidelines to condemn the measurement process this approach provides flexibility based on knowledge The precision to tolerance ratio The P T ratio was originally created in an attempt to describe how much of the specified tolerance was consumed by measurement error While this ratio is quite simple it fails to do what was intended Originally the precision to tolerance ratio was defined using a numerator of 515 times the standard deviation of measurement error or 515 sigma e The multiplier of 515 was obtained from the width of the interval that covers the middle 99 percent of a standard normal distribution Thus the idea was to filter out 99 percent of the contribution of measurement error Around 1990 the Automotive Industry Action Group AIAG changed the multiplier from 515 to 600 and this is the version of the P T ratio found in most software today For an example consider a physical dimension that has specifications of 3006 inches to 3012 inches Say that the measurement device records this dimension to the nearest 0001 inch so that the measurement increment MI is one-thousandth of an inch Figure 1 Specifications are usually expressed as possible values Then if measurement error is sigma e 00007 inches the P T ratio will be 06000 or 60 percent which is a very bad number indeed Dont write to us if your computer gives you something different it is probably not programmed to correctly compute the specified tolerance For more on this see figure 2 Regardless of whether we use either 515 or 60 in the numerator the P T ratio will overstate the impact of measurement error This happens because neither version of this ratio has the appropriate mathematical foundation As a result neither version computes the consumption in the proper manner Since guard-bands based on the traditional P T ratio will end up rejecting an excess amount of good product all producers should be interested in computing appropriate manufacturing specifications Guard bands and manufacturing specifications Like the P T ratio most guard bands are based on a probability of getting a conforming measurement an outcome given that the item is nonconforming a state of nature While conditional probabilities of an outcome given a state of nature are inputs to the problem they are not the results needed to properly construct guard-bands What is needed are the a posteriori probabilities the probability of a state of nature given an outcome the probability of a conforming item given that the measurement falls within the manufacturing specifications This necessity of reversing the form of the probabilities to answer practical questions is one of the fundamental laws of probability theory that has been known since the 18th century The mathematics behind computing the a posteriori probabilities needed for defining appropriate guard-bands for manufacturing specifications are outlined in my article Where do Manufacturing Specifications Come From Quality Digest Daily July 6 2010 and they are given in greater detail in Chapter 14 of my book EMP III Evaluating the Measurement Process and Using Imperfect Data SPC Press 2006 These guard bands are characterized by the minimum probability that the product is conforming a state of nature when the measurement falls within the manufacturing specifications an observed outcome We shall use this approach to discuss three different sets of manufacturing specifications For these three sets of guard bands the Honest P T ratios will be respectively 225 percent 45 percent and 675 percent as large as the traditional P T ratio 85 manufacturing specifications If you want the probability of conforming product to be at least 85 percent then you will need to tighten the watershed specifications by guard bands of GB 85 measurement units on each end where If A and B denote the smallest and largest acceptable values and if MI represents the measurement increment the watershed specifications will be Figure 2 Watershed specifications fall between the possible values The 85 percent manufacturing specifications will be Items with measurement values that fall within these 85 percent manufacturing specifications will have at least an 85-percent likelihood of conforming to the customer specifications The proportion of the specified tolerance that is consumed by these 85 percent guard bands is characterized by the Honest P T 85 ratio This Honest P T 85 ratio will be 225 percent as large as the original P T ratio when that value has been computed correctly using the watershed specifications For our example A 3006 B 3012 MI 0001 sigma e 00007 and the traditional P T ratio is 060 The guard band for 85 percent manufacturing specifications would be So the 85 percent manufacturing specifications are And the possible values within these specifications are 3006 inches to 3012 inches When a measurement falls within this range the item has at least an 85-percent chance of being conforming Here the Honest P T 85 ratio is 0134 which describes that proportion of the specified tolerance 134 that is consumed by the 85 percent guard bands 96 manufacturing specifications If you want the probability of conforming product to be at least 96 percent then you will need to tighten the specifications by guard bands of GB 96 measurement units on each end where and the 96 percent manufacturing specifications will be Items with measurement values that fall within these 96 percent manufacturing specifications will have at least an 96-percent likelihood of conforming to the customer specifications The proportion of the specified tolerance that is consumed by these 96 percent guard bands is characterized by the Honest P T 96 ratio This Honest P T 96 ratio will be 45 percent as large as the original P T ratio For our example A 3006 B 3012 MI 0001 sigma e 00007 and the traditional P T ratio is 060 The guard bands for 96 manufacturing specifications would be Here our 96 percent manufacturing specifications are And the possible values that fall within these specs are 3007 inches to 3011 inches When an item gets a measurement within this range it has at least an 96-percent chance of being conforming Here the Honest P T 96 ratio is 0270 which describes that proportion of the specified tolerance that is consumed by the 96 guard bands 99 manufacturing specifications If you want the probability of conforming product to be at least 99 percent then you will need to tighten the specifications by guard bands of GB 99 measurement units on each end where and the 99 percent manufacturing specifications will be Items with measurement values that fall within these 99 percent manufacturing specifications will have at least an 99-percent likelihood of conforming to the customer specifications The proportion of the specified tolerance that is consumed by these 99 percent guard bands is characterized by the Honest P T 99 ratio This Honest P T 99 ratio will be 675 percent as large as the original P T ratio For our example A 3006 B 3012 MI 0001 sigma e 00007 and the traditional P T ratio is 060 The guard bands for 99 percent manufacturing specifications would be Here our 99 percent manufacturing specifications are And the possible values that fall within these specs are 3007 inches to 3011 inches Thus due to the round-off of the measurement increment in this case the 96 percent and 99 percent manufacturing specifications turn out to include the same range of possible values When an item gets a measurement within this range it has at least an 99-percent chance of being conforming The Honest P T 99 ratio is 0405 which describes that proportion of the specified tolerance that is consumed by the 99 percent guard bands This rigorous approach to the definition of guard bands gives you a choice between three different sets of manufacturing specifications having different minimum probabilities of conforming product of 85 percent 96 percent or 99 percent In this case the guard bands for these manufacturing specifications consume 135 percent 27 percent and 405 percent of the specified tolerance respectively The traditional P T ratio claims that measurement error consumes 60 percent of the specified tolerance Thus the inflation that is inherent in traditional P T ratio plus the conservative nature of the traditional guideline for interpreting the P T ratio combine to effectively condemn most measurement systems On the other hand correctly computed guard bands and their Honest P T ratios give us options for using the current measurement system Since we always have to work with imperfect measurement systems these options are important If we condemn a measurement system then that system will have to be replaced This will require time effort and capital expenditures Since expenditures on measurement systems are always an overhead expense they should not be undertaken unnecessarily If we do not condemn the measurement system but use guard bands based on the original P T ratio then the tightened specifications will be narrower than they should be and an excess amount of good product will be rejected unnecessarily The use of inflated P T ratios and arbitrarily conservative guidelines may allow us to beat our vendors over the head but it does nothing to increase quality productivity or competitive position Understanding how to work within the limitations of the current measurement system allows us to avoid unnecessary costs

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