Estimating Gage Repeatability using Range Statistics for one operator is not all that difficult. Is the same true when estimating Gage Repeatability using Range Statistics when using two operators? Let’s discuss this question in more detail…
In a previous post we looked at estimating Gage Repeatability using Range Statistics. In that post we examined one operator measuring ten parts twice. In this post we’ll estimate Gage Repeatability using the same ten parts measured by two operators.
The data set used, for both operators, appears below. In each table shown are two values for the same part feature. To compute the average range we must first calculate the range for each part by operator. I also show this in the tables below. Next, we need to compute the average range. To do so we must add all the range values, by operator, and divide by the number of parts measured. In this case, each operator measured 10 parts.
The Average range for operator, 1 and 2, is 3.8 and 3.6. In statistics, we often place a horizontal bar above a variable to denote an average. In this case we will refer to each average range as Rbar1 and Rbar2. Now that we have the average range for each operator, we need to compute a single average range. To do so, we add the average range by operator and divide by 2. This yields an average range of 3.7 ( [ 3.8 + 3.6 ] / 2). We call this R-double bar since we computed an average from an average. We use two horizontal bars above the Range statistic to denote this.
Now that we have computed an average range across both operators we can use this value to estimate the standard deviation. To do so we need to use a d2 correction factor when estimating the standard deviation using a range statistic. The d2 correction factor depends on the number replicate readings (r). For our example, each operator measured the same feature on the same part twice. As such, there are r=2 replicate values for each part. The d2 values for r=2 and r=3 replicate values is 1.128 and 1.693. For those of you that have an interest in knowing where d2 values come from and how to compute them then read this post.
In our example, r=2 so we will use d2=1.128. Using this value we can compute the standard deviation using the following expression.
Dividing d2 into R-double bar we have an estimate of the standard deviation due to gage repeatability. The standard deviation due to Gage Repeatability is 3.28.
But our analysis does not end here. We need to check that the standard deviation due to Gage Repeatability is a reasonable estimate.
Recall that each range is the absolute value of the difference between two values. As such, we must check to see that these range values are reasonable using a Range chart. To do so we must compute the Upper Control Limit (UCL) for the ranges. To compute the UCL for the ranges we use the following expression.
In this expression, D4 is a constant that is a function of the number of replicate values per part. In this case, we had r=2 replicate values for each part. The table below shows the D4 values for r=2 and r=3 replicate values per part. The UCL for the ranges is 12.09.
Any range value that exceeds the UCL for the ranges suggests an excessive range value for a specific part. In this case, all the range value falls below the UCL for the ranges. If any of the range values exceeded UCL then verify that the data was not entered in error. If not entered in error, then have the same operator re-measure the part feature again and re-compute the range. If the new range value falls below the UCL for the ranges then use that value.
There is one more step we can take to assure R-double bar is reasonable. This step isn’t discussed in the 4th edition MSA manual published by the AIAG. But, I believe it is important.
Recall R-double bar is an average of the average range by operator. When we compute R-double bar we assume that the average range by operator is similar. When one average range is larger than another, then R-double bar becomes inflated. Since we use R-double bar to computer the UCL for the range it also becomes inflated. We can test if the average range by operator is different by applying the following F-test using the Range.
This F-test is only applicable when the number of replicate values per part (r) and the number of parts (k) measured by operator are equal. In our example, each operator obtained r=2 replicate values per part and they each measured k=10 parts. To compute an F-value, place the larger and smaller average range in the numerator and denominator then square the average range to get an F-value. I show this calculation below.
Once we compute an F-value we need to compare it to a critical F-value. We can find critical F-values in an F-table. These values are a function of the number of degrees of freedom (df) used to estimate the average range. Since standard practice is to use k = 10 parts and r=2 or r=3 replicates the degrees of freedom for each case appears in the table below.
In our example we have df = 9. With this information we can find the critical F-value assuming a 5% risk. Since we don’t know which average range is larger we have two choices to choose from when performing an F-test. As such, we split our 5% (0.05) risk between both possibilities and use 2.5% (0.025) to get a critical F-value. The critical F-value, found in an F-table, appears below.
If the computed F-value < F Table (critical) there is no evidence to suggest the Average Range by operator is different. In this case, 1.11 < 6.33 thus R-double bar is a reasonable estimate. In our example, we can estimated Gage Repeatability at 3.28 (R-double bar / d2 = 3.7 / 1.128 = 3.28).
In summary, I like to check the Average Range by operator using an F-test. If there is no evidence to suggest they are different, computing a single Average Range is reasonable. As a final check, I like to compute the UCL for ranges and see if any part feature had a range value that exceeds the UCL. If a range value exceeds the UCL, I then check if an operator entered the values in error. If entered in error I have the operators re-measure the part feature and check if the new range value falls below the UCL.
Now it’s your turn to let me know if you enjoyed this post or if you have any questions. Please leave your comments below. I look forward to them.