Manuscript No. 189
A revid version of the paper read at 2006 ASQ/ASA Fall Technical Conference 1
中国十大银饰品牌An Honest Gauge R&R Study Donald J. Wheeler January 2009
In a 1994 paper Stanley Deming reported that when he did his literature arch on mea-surement error he had expected to find a single, original paper followed by many extensions and additions in a logical stream of development. Instead, he found not one stream but many different rivulets, each developed without reference to each other, and most of which were done without the aid of any mathematical rationale or support. In short, the problem of mea-surement error is so widely recognized that many different solutions, in different fields of endeavor, have been propod. The various solutions cover the spectrum from naive to theoretical, from simple to complex, and from wrong to right.
One such solution that has been widely promoted in the automotive world and beyond is known as a Gauge R&R Study. While this technique can be traced back to at least 1962, over the years it has been subject to many generations of changes. After being subjected to rious revisions over the past 20 years some of the more egregious mistakes have been eliminated.However, fundamental problems remain. This paper will illustrate the problems and pro-po an alternative that is both simple and co
rrect.1. The AIAG Gauge R&R Study
An example will be helpful in our discussion of a Gauge R&R Study. Generally a Gauge R&R Study will have two or more operators, one gauge, and up to ten parts. Each operator will then measure each part two or three times, resulting in a fully-crosd data structure with subgroups of size two or three. For our purpos let us say that we have three operators who will u a micrometer to measure the thickness of each of five gaskets twice. Thus, this study has n = 2, o = 3, and p = 5. The 30 data (in mils) are shown in Table 1.
Table 1 The Gasket Thickness Data
Operator A B C Part 1234512345123451st Value 1672101871891561552061821841431522061801801462nd Value 162213183196147157199179178142155203181182154Averages 164.5211.5185.0192.5151.5156.0202.5180.5181.0142.5153.5204.5180.5181.0150.0Ranges 534792736133128
An Honest Gauge R&R Study
2
Since 1990 the Gauge R&R Study has been tended by the Auto Industry Action Group of the American Society for Quality. References to the current version refer to the Gauge R&R Study as it is prented and explained in the Third Edition of the AIAG’s Measurement System Analysis. The following is a step-by-step discussion of the current version.
Step 1. The AIAG Gauge R&R Study begins by calculating the Average Range for the k = 15
subgroups of size n = 2 of Table 1 and finding the corresponding Upper Range Limit.Then the obrved ranges are compared to this Upper Range Limit to e if any exceed this limit. For Table 1 the Average Range is 4.267 mils and the URL = 13.9 mils. None of the ranges in Table 1 exceed this limit.
Step 2. Next the Average Range from Step 1 is divided by the appropriate Bias Correction
Factor to obtain an estimate of the standard deviation for measurement error. This esti-mate is called the Repeatability or the Equipment Variation (EV ).
EV = ^σpe = R –d 2 = 4.2671.128
= 3.783 mils Step 3. Next the Reproducibility or Apprair Variation (AV ) is estimated. The formula
ud here has evolved over the years. The current version us the range of the Operator Averages. For our example the o = 3 Operator Averages are 181.0, 172.5, and 173.9. The range of the three averages is R o = 8.5 mils. The Bias Correction Factor for ranges ud here is the Bias Correction Factor for estimating variances which is commonly known as d 2*. For the range of three values this value is d 2* = 1.906, and each of the Operator Aver-ages was bad upon [(n o p) / o ] = 10 original data. The formula for the Reproducibility is:
AV = ^σo = √ [ R o 2]2
河北省旅游景点– o ^σpe 2 = √ [ 8.5]2 – 3 3.7832
= 4.296 mils Step 4. Next the Combined Repeatability and Reproducibility (Gage R&R) is found by
squaring the results of Steps 2 and 3, adding them together, and taking the square root:
GRR = ^σe = √ EV 2 + AV 2 = √
3.7832 +
4.2962 =
5.724 mils Step 5. Next the Product Variation (PV ) is estimated using the range of the p = 5 Part
Averages. The Part Averages are 158.0, 206.167, 182.0, 184.833, and 148.0. Their range is R p = 58.167. Once again the Bias Correction Factor for the range of five values is the one for estimating variances, d 2* = 2.477. Thus the Product Variation is estimated to be:
PV = ^σp = R p d 2* = 58.1672.477 = 23.483 mils Step 6. Finally the Total Variation (TV ) is estimated by combining the Product Variation
with the Repeatability and the Reproducibility to get:
TV = ^σx = √ EV 2 + AV 2 + PV 2 = √
3.7832 +
4.2962 + 23.4832 = 24.171 mils
Donald J. Wheeler Up to this point everything is okay. While the estimators defined here are not the only estimators that could have been ud, and while they are not always unbiad estimators, they do provide reasonable estimates for the quantities described. In fact, the estimator in Step 2 is Esti
mator 1 from Table 4. The estimator in Step 3 is the square root of Estimator 11 from Table 4, and the estimator in Step 5 is the square root of E stimator 6 from Table 4. (While the AIAG Gauge R&R computational worksheet simplifies the formulas by defining veral “K-factors” the AIAG formulas are algebraically equivalent to tho given on the pre-ceding page.)
The train wreck begins when the AIAG Gauge R&R Study tries to u the estimates from the preceding steps to characterize relative utility. In the current version the quantities from Steps 2, 3, 4, and 5 are all expresd as a percentage of the last value, the Total Variation. 2. The “Percentages” of the Total Variation
Step 7. The Repeatability (EV) is divided by the Total Variation (TV) and multiplied by 100 to be expresd as a percentage. This ratio is labeled %EV and is interpreted as the per-centage of the Total Variation that is consumed by Repeatability or Equipment Variation.
For the data of Table 1 this computation will yield a value of:
%EV = 100 [ 3.783/24.171 ] = 15.65%
Step 8. T he Reproducibility (AV) is divided by the Total Variation (TV) and multiplied by 100 to get a
value denoted by %AV that is interpreted as that percentage of the Total Variation that is consumed by Reproducibility. For the data of Table 1 this computation will yield a value of:
%AV = 100 [ 4.296/24.171 ] = 17.77%
南京珍珠泉Step 9. The Combined Repeatability and Reproducibility (GRR) is divided by the Total Vari-ation and multiplied by 100 to get a value denoted by %GRR that is interpreted as that percentage of the Total Variation that is consumed by Combined R&R. For the data of Table 1 this computation will yield a value of:纪念日礼物
%GRR = 100 [ 5.724/24.171 ] = 23.68%
Step 10. The Product Variation is divided by the Total Variation and multiplied by 100 to get
a value denoted by %PV that is interpreted as that percentage of the Total Variation that
is consumed by the Product Variation. For the data of Table 1 this computation will yield
a value of:
%PV = 100 [ 23.483/24.171 ] = 97.15%
未雨绸缪的故事
Following the formulas in the current version of the AIAG manual there is a simple statement to the effect the “The sum of the percent consumed by each factor will not equal 100%.” This statement has no explanation attached. There is no guidance offered on how to
3
An Honest Gauge R&R Study
4proceed now that common n and every rule in arithmetic have been violated. Just a simple statement that the numbers do not mean what they were just interpreted to mean,and the ur is left to his or her own devices. Unfortunately, unlike the Red Queen in Won-derland, when it comes to arithmetic we do not get to say that things mean whatever we want them to mean.
WHY THESE “PERCENTAGES” DO NOT ADD UP
The %EV and %AV do not add up to the %GRR becau they are not proportions. Like-wi, the %GRR and the %PV do not add up to 100% becau they are not proportions. They are instead trigonometric functions. Figure 1 shows how the five estimates found in Step 2through Step 6 are related.
Figure 1: The Quantities Ud for the Ratios in Steps 7, 8, 9, & 10
朝花夕拾读书笔记摘抄
Before multiplication by 100, the ratio computed in Step 7 can be expresd in terms of Angles A and B as:
%EV 100 = (Sine A) (Cosine B) = 5.72424.171 3.7835.724 = 0.1565Before multiplication by 100, the ratio computed in Step 8 can be expresd in terms of Angles A and B as:%AV = (Sine A) (Sine B) = 5.724 4.296 = 0.1777
Before multiplication by 100, the ratio computed in Step 9 can be expresd in terms of Angle
A as:%GRR 100 = (Sine A) = 5.72424.171 = 0.2368
Before multiplication by 100, the ratio computed in Step 10 can be expresd in terms of Angle A as:%PV = (Cosine A) = 23.483 = 0.9715
绽放青春In this form we can begin to e why the quantities do not add up. While they were dresd up to look like proportions, and while they were interpreted as proportions, they are, and always have been, nothing more than trigonometric functions. And trigonometric functions do not satisfy the conditions needed for a t of ratios to be interpreted as proportions.
Donald J. Wheeler
5a + b + c = 1A t of ratios are proportions only when the denominator is the sum of the numerators.This additivity of the numerators is the esnce of proportions. So what is additive in a Gauge R&R Study? Look at the structure of the formula in Step 6. The Total Variance is the sum of the Repeatability Variance , the Reproducibility Variance , and the Product Variance .
σx 2 = σpe 2 + σo 2 + σp 2
Since the variances are additive, we know from the Pythagorean Theorem that the standard deviations cannot be additive . However, when the ratios of Steps 7, 8, 9, and 10 are expresd as percentages and interpreted as proportions they implicitly assume that the standard deviations are additive. This implicit assumption of additivity is a violation of the Pythagorean Theorem, and is what makes it impossible to make n of the ratios in Steps 7,8, 9, and 10. It is why the “percentages”
do not add up, and this is why engineers have told me they never could figure out exactly what the final numbers in a Gauge R&R Study repre-nted. They sound like nonn becau they are trigonometric functions being interpreted as proportions when they are not proportions.
SO WHAT ARE THE ACTUAL PROPORTIONS?
What proportion of the Total Variation is attributable to Repeatability or E quipment Variation? The equations above suggest that a reasonable estimate would be:
^σpe 2 ^σ
有分x 2 = EV 2TV 2 = 3.7832
24.1712 = 0.0245or 2.45% rather than the 15.65% erroneously found in Step 7.
What proportion of the Total Variation is attributable to Reproducibility or Apprair Variation? The equations above suggest that a reasonable estimate would be:
^σo 2 ^σ
x 2 = AV 22 = 4.29622 = 0.0316or 3.16% rather than the 17.77% erroneously found in Step 8.
What proportion of the Total Variation is attributable to Combined R&R? The equations above suggests that a reasonable estimate would be:
^σe 2 ^σ
x 2 = GRR 2TV 2 = 5.7242
24.1712 = 0.0561or 5.61% rather than the 23.68% erroneously found in Step 9. Note that this value is the sum of the two preceding values.
What proportion of the Total Variation is attributable to Product Variation? The equa-tions above suggest that a reasonable estimate would be:
