Performing Gage Repeatability & Reproducibility (GR&R) Studies On Automated Test Equipment

In Practical Terms,

Some Measurement Errors When Considering Selling and Purchasing Potato Salad

by John Raffaldi

Gage Repeatability & Reproducibility (GR&R) studies focus on identifying measurement variation caused by the gages used for measurements, and people who perform the measurements, more commonly known as appraisers. Using Gage Repeatability & Reproducibility studies, it’s possible to identify measurement variation and determine if the measurement variation is too large for the intended use. DATA GATHERING STEPS Select 10 items (weights of potato salad containers) which cover the range of the items that will be measured.Use only one measurement system (in our case a scale). Each item is measured (in our cased weighed) two to 10 times depending on how much time you have and the measurements (in our case weights) are recorded. Perform the calculations (which may be used in a spreadsheet application) are as follows.
Some measurement equipment does not neatly fit into typical Gage Repeatability & Reproducibility study methods because there is no variation caused by the people doing the measurement. An example of this would be using a scale to weigh objects. For automated measurement equipment such as a scale, only the measurement error caused by the equipment, known as Equipment Variation (EV) must be calculated for a Gage Repeatability & Reproducibility study. As explained above, the variation caused by the people performing the measurement, usually called Appraiser Variation (AV), does not exist. In other words, what the scale reads for a measurement does not depend on how the object is placed on the scale. This is in contrast to a measurement with an instrument such as a micrometer where measurement technique (such as spindle contact feel) can cause differences between the readings people obtain. The following method for Gage Repeatability & Reproducibility studies on automated test equipment has found wide use in the semiconductor industry and will work well for this example. Automated test equipment is found all over, and is not limited to industry and does not have to be expensive. When you use your bathroom scale, you are using a piece of automated test equipment. So is the person in back of the deli counter using a piece of automated test equipment that weighs a pound of potato salad and calculates the cost. Suppose we have a product we sell in different weights such as pre-filled containers of potato salad like found at the supermarket, and we would like to know the measurement error produced by our calibrated scale relative to a tolerance. The following provides an overview and example of how to perform the study. CALCULATIONS STEPS We will use information about the weights in the calculation example later, but here are the steps: Calculate the standard deviation from the trials for each part. Calculate the average standard deviation, sBar from the step above. Divide by c4 from Duncan in Table M. The number of observations in the sample, n, is the number of trials. Multiply by the number of sigma you want to use, usually six. Divide by the tolerance and multiplying by 100 to obtain the P/T as a percentage of the tolerance.
The formula from the steps above is: Gage Repeatability & Reproducibility as a percentage of the tolerance = ((6 * sBar / c4) / (USL - LSL))*100 sBar = The sum of the standard deviations for each part measured divided by the number of packages of weights used in the study. c4 values from the text book Duncan, Table M (Quality Control and Industrial Statistics, Duncan, Acheson J., Irwin Inc., 1986)
Trials	c4 value
2	0.7979
3	0.8862
4	0.9213
5	0.9400
6	0.9515
7	0.9594
8	0.9650
9	0.9693
10	0.9727

AN EXAMPLE

Suppose we have 10 containers of potato salad which span the range of what our customers typically purchase, one to ten pounds and we want our product’s package weight to be off no more than +/- .0025 lbs. We would like to know as a percentage of our allowable weight variation, the tolerance, how much measurement error is produced by the scale. The importance in finding this out is that if our measurement error produced by the scales is to large is that we will frequently make mistakes saying the container weight is within the allowable weight range when it truly is not, short changing our customers if we are at a weight near the -.0025 lower weight variation. Likewise, we will be loosing money if we say the container is under weight when it is not, we will add more product. Other ideas about scale accuracy are important, but we will ignore them for this example. As shown below, instead of taking a measurement like 7.0143 pounds for the 7-pound potato salad container, we will report only the .0143, the decimals, because we are only interested in the variation caused by the scale.

The weights we measure three times, also known as three trials are as follows:

Package Size In Pounds	TRIAL 1 Variation	TRIAL 2 Variation	TRIAL 3 Variation
1	-0.0593	-0.0591	-0.0596
2	-0.0384	-0.0386	-0.0381
3	0.0023	0.0022	0.0024
4	-0.0280	-0.0282	-0.0281
5	0.0012	0.0012	0.0010
6	0.0225	0.0221	0.0223
7	0.0143	0.0142	0.0141
8	0.0008	0.0009	0.0006
9	0.0336	0.0335	0.0332
10	0.0741	0.0743	0.0744

Calculating the average standard deviation as described above using a spreadsheet application we have:

Package Size In PoundsTRIAL 1TRIAL 2TRIAL 3 STD DEV
1-0.0593 -0.0591 -0.0596 0.00025166
2-0.0384 -0.0386 -0.0381 0.00025166
30.0023 0.0022 0.0024 0.00010000
4-0.0280 -0.0282 -0.0281 0.00010000
50.0012 0.0012 0.0010 0.00011547
60.0225 0.0221 0.0223 0.00020000
70.0143 0.0142 0.0141 0.00010000
80.0008 0.0009 0.0006 0.00015275
90.0336 0.0335 0.0332 0.00020817
100.0741 0.0743 0.0744 0.00015275
AVG STD DEV	0.00016325

Using the equation:

Gage Repeatability & Reproducibility as a percentage of the tolerance = ((6 * sBar / c4) / (USL - LSL))*100 With an Upper Specification Limit (USL) of .0025 indicating the weight can be .0025 pounds above the target weight and be in tolerance and a Lower Specification Limit (LSL) of -.0025 indicating a container can not be less than .0025 pounds below the target weight, and a c4 value of 0.8862 corresponding to three trials we have: Gage Repeatability & Reproducibility as a percentage of the tolerance = ((6 * 0.00016 / 0.8862) / (.0025 - (-.0025))*100 Resulting in: Gage Repeatability & Reproducibility as a percentage of the tolerance = 22.1% Which indicates that a little more that a fifth of the allowable range of +/-.0025 pounds is lost to measurement error. This is higher than is generally accepted for measurement error indicating the scale may be unsatisfactory. Another way of looking at it is to say, for a container that weighs within the +/- .0025 pounds, but very close to either the + .0025 or -.0025 acceptable range of variation has a 1 in 5 chance of being out of being outside the acceptable range even though it is truly within the acceptable range.

Likewise a container that is too high or low in weight near one of the specifications has a 1 in 5 chance of being measured within the proper weight although it is truly beyond the +/-.0025 weight difference than we can accept.