## Gage R&R and Process Variation

This is the second in a four-part series on Gage R&R. The first blog explained what a Gage R&R study is. This blog examines the relationship between the Gage R&R results and the process variation and answers the question:

*Is the measurement system capable of telling the difference between the parts or samples taken the process?*

This is just another way of asking if the measurement system can be used to control the process. The basic equation describing the relationship between the total variance, the product variance and the measurement system variance is given below.

σ_{x}^{2} = σ_{p}^{2}+σ_{e}^{2}

where σ_{x}^{2} is the product measurements variance, σ_{p}^{2} is the product variance, and σ_{e}^{2} is the measurement system variance.

In a Gage R&R study, each operator measures each part multiple times (this gives the repeatability, the first R) and the operator’s average measurement for each part is compared to the average of the other operators (this gives reproducibility, the second R). Consider the output from a Gage R&R study as shown below. The first column is the source of variation. The second column is the variance associated with that source. The third column gives the % of the total variance for each source (its contribution to the total variance). You can see the data for this example in our SPC Knowledge Base article titled "Acceptance Criteria for Measurement Systems Analysis (MSA)".

Source | Variance | % Contribution |
---|---|---|

Total Gage R&R | 47.72 | 8.44% |

Repeatability | 31.64 | 5.60% |

Reproducibility | 16.08 | 2.84% |

Part-to-Part | 517.7 | 91.56% |

Total Variation | 565.4 | 100.00% |

The “Total Gage R&R” in the above table is the combined variance due to repeatability and reproducibility. This is the σ_{e}^{2} term in the equation above. Likewise, the “Part-to-Part” variance corresponds to the σ_{p}^{2} term. And finally, the “Total Variance” corresponds to the σ_{x}^{2} term.

A Gage R&R then should estimate these variances because that allows you to determine the % of the total variance due to the measurement system:

% of Total Variance due to Measurement System = 100(σ_{e}^{2}/σ_{x}^{2})

This number gives you an idea of “how good” your measurement system is at telling the difference between parts from your process. In the example above, the % of total variance due to the measurement system is 8.44%. Obviously, smaller numbers are better – the more the measurement system can distinguish between parts. Too large of numbers and all you see is measurement error, not the differences in the parts.

Are there guidelines for what the % of the total variance due to the measurement system should be? Well, there are. But you need to be careful about how you interpret those guidelines. For % of the total variance due to the measurement system, one set of guidelines is:

- Under 1%: generally considered to be an adequate measurement system
- 1 to 9%: may be acceptable for some applications
- Over 10%: considered to be unacceptable

There does not appear to be any logic for these values that I have seen. They are related to the AIAG (American Automotive Action Industry) guidelines for their average/range Gage R&R method. But again, there is no logic to those numbers. A better method is needed. And that will be the subject of our next blog.

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