My Knowledge in Quality Management: How to Calculate GRR%

For GRR analysis on a measurement system, GRR% is used to determine whether the repeatability and reproducibility of the measurement system is acceptable for the purpose of measurement. It is calculated with Equation 1:

In the AIAG’s MSA manual (4th edition), four different approaches are presented to determine TV based on different scenarios:

Accordingly, TV is calculated with the following equations:

The above four approaches and the corresponding equations for TV may be confusing to many people who are in charge of MSA in their companies, and they may not be really clear of which equation should be used in actual cases they encounter when doing their GRR analysis. This article is intended to give some additional discussion and clarification about how to decide which equation should be used for TV and in turn GRR% calculation.

To decide which equation to be used, one first needs to consider the purpose of measurement. In general, any measurement conducted on a process has two purposes:

To judge whether the measured parameter is within or out of the spec limit, for the purpose of OK/NG judgement or product control;
To decide whether the process is under control (without special cause) and whether the process capability (Cp and Cpk) and performance (Pp and Ppk) meet the requirements, i.e. for the purpose of statistical process control (SPC).

1. When the measurement is for product control

When the measurement is for product control, one cares about the possibility of misjudgment. Misjudgment happens when the true value of a measured parameter is located near the spec limit, indicated with the gray region in the graph below.

The width of the gray region is determined by the variation resulted from the measurement system, i.e. the GRR of the measurement system (The width of gray region is 6 times of GRR. Outside this width, the possibility of misjudgment due to GRR is nearly 0). The wider the gray region is in respective of the tolerance (USL-LSL), the higher the possibility of misjudgment is. Therefore, in order to have a small possibility of misjudgment so that the measurement can really achieve the purpose of product control, the ratio between gray region and the tolerance 6*GRR/(USL-LSL) must be small enough. GRR% should hence be evaluated with the following equation, i.e. the 4th approach presented in the MSA manual:

2. When the measurement is for process control

When the measurement is for process control, one cares whether the obtained data can correctly identify the existence of special causes and whether the measurement can really reveal the true process capability (Cp and Cpk) and performance (Pp and Ppk). Let’s take a look at these two purposes respectively.

2-1. When the measurement is to identify the existence of special causes

In AIAG’s SPC manual, it listed out 8 criteria to identify the existence of special causes, as shown below:

Among all the 8 criteria, the 1st one is mandatory, and the rest are optional. So the discussion here will be just based on the 1st criterion.

To identify the special causes with the 1st criterion, one needs to monitor whether there’s any data point out of the control limits (±3σp). It is similar to judge NG samples from good ones. The difference is that the former uses the control limits, while the latter uses the spec limits. To correctly identify the existence of special causes, one cares about the possibility of misjudgment (i.e. an under-control data point is misjudged as an out-of-control data point, or an out-of-control data point is misjudged as an under-control data point). This is determined by the ratio of the gray region around the control limits with respective to the width of the control limits, i.e. 6*GRR/(UCL-LCL). Therefore, GRR% should be evaluated with the following equation:

where σp is the process variation.

2-2. When measurement is to monitor the process capability or process performance

For this purpose, let’s just see how GRR of a measurement system impacts the observed Ppk of a process. It’s similar for other index, including Cpk, Cp and Pp.

Ppk of a process is calculated with the following equation (please refer to P.133 of the 2nd edition of AIAG’s SPC manual)

where σp is the total process variation as observed from the measurement results. This observed process variation includes two sources: one is the actual variation of the process itself σ1, and the other is the variation from the measurement system, which is GRR. According to the property of the normal distribution

So Equation 8 can be rewritten as

It should be noted that Ppk as calculated above is the observed Ppk by the measuring process. It does not reflect the actual performance of the process. The actual Ppk should be calculated as below, without the variation from the measurement system in the denominator:

Comparing Equation 10 with Equation 11, one can see that the larger GRR is, the bigger the difference is between the observed Ppk and the actual Ppk (the former is always smaller). In order to make process control effective, the observed Ppk must be as close as to the actual Ppk, so it causes no misjudgment on the actual process performance. In other words, GRR must be significantly smaller compared with σp, so that Equation 11 can be approximated with Equation 8. Therefore GRR% of the measurement system should be evaluated with the following equation:

This equation is equivalent to Equation 7. So for either purpose 2-1 or 2-2, it comes to the same conclusion, i.e. GRR% should be evaluated as the ratio of GRR with respect to the process variation.

The process variation can be known or unknown when GRR analysis is conducted. Let’s now see how to calculate GRR% based on Equation 7 or 12 in both cases.

When the process variation is unknown, for example, GRR analysis is done when the process is still in design and development stage or the process has just undergone some changes and it has not been studied yet. In such a case, the observed process variation σp can be obtained with following approaches:

Select a group of samples from the process (It must be noted that the number of parts must be enough to cover the full width of the process variation) and measure the parts variation PV in this group. PV represents the true variation of the process, so the observed process variation σp, according Equation 9, is:

GRR%, hence, can be calculated with the following equation, i.e. the 1st approach introduced in MSA’s manual:
Use the process variation σp of a similar process, i.e. the 2nd approach:

This approach should only be used when it’s not possible to obtain and do measurement on a group of samples representing the full process variation, i.e. the 1st approach is not feasible.
Use the target Pp or Ppk, i.e. the 3rd approach:

Again, this approach should be used only if the 1st approach is not feasible.

When GRR analysis is done when the process has already been studied and released to mass production, for example, the annual MSA after process release, the process variation is either directly known or can be derived from the other process index, and GRR% can be calculated with the following approaches:

The process variation is directly known, GRR% can be calculated with the 2nd approach.
When process variation is not known, but Pp is known, GRR% can be calculated with the 3rd approach.
When the process variation is not known, but the variation of Xbar, i.e. or control limits of Xbar, i.e.is known, GRR% can be calculated with the following equation:

This approach is not mentioned in the MSA manual
If a group of samples is available which represents the full width of the process variation, GRR% can also be calculated with the 1st approach, without using the existing data about the process.