In this series of posts of Gage R&R – Six sigma tool, I review the purpose, use, interpretation, and limitations of various six sigma tools.

This post is about Gage R&R, a critical tool in the Measure phase and the Control phase of DMAIC. There are many aspects of Gage Repeatability and Reproduceability. I'll attempt to cover the major points here, starting with the purpose(s).

### Overview

When a decision is made to charter a six sigma project. One of the major concerns early into the project is the reliability of the data that is use to determine root causes. It is very important that the data that is analyze about the project problem be accurate. Consistent to enable accurate analysis, and that good conclusions can be drawn from the data.

*Purpose for Gage R&R*

This is important to ensure that improvement plans are effective at addressing the root causes. It can deliver real improvement. Gage R&R also comes up again in the Control Phase of DMAIC to help ensure that the significant parts of the improvement plan can be accurately measure.

There is another purpose for Gage R&R. Actually Gage R&R is primarily a tool for determining if measurement systems used to evaluate the quality aspects of a product will produce reliable results. In either situation, the intent of Gage R&R is to give an indication of the proportion of the variation that is present in our system that comes from the measurement system.

At its very basic level, a measurement system must be able to distinguish good product from bad product. Understanding the ability of the measurement system to do that is the purpose of a Gage R&R study.

*Aspects to a Gauge R&R study*

There are several aspects to a Gage R&R study. Among the most important things to consider are:

- Reproduceability
- Repeatability
- Accuracy
- Precision
- Bias
- Linearity
- Sample Selection

*Lets start with Reproduceability*

Reproduceability is the measurement of the portion of the variation in the measurement system that is coming from differences between people. Most measurement systems consist of two primary components of variation; People induced variation, and Instrument induced variation. Reproduceability tells us about differences between the ways that people do the steps of a measurement method and how much those differences matter.

Repeatability is the portion of the measurement system variation that comes from the instrument itself. Repeatability is a measure of the ability of the measurement device to deliver a consistent result over several measurements.

*Accuracy and Precision*

The accuracy and Precision can be taken together. An accuracy is a measurement of the measured result compare to the true result. Remember that our measurement system is intent to give us a high confidence in the data. That we use to decide on product quality or determine the root causes and improvement plans for six sigma. Precision is a measurement of the variation in results seen. Think of these two like a bulls eye target.

See below for a visual example showing the relationship between Accuracy and Precision.

*Conclusion from Visual tool*

Bias is the difference between the measure result of a sample and the actual result of that same sample. Bias is the error that exists in the measurement system. In the example below we are looking at our car speedometer.

If we compare the measured result of the speedometer at three speeds (30, 50, and 70 mph) and we can know the actual speed that the car is going, we can determine the bias or error across the range of interest of the measurement system.

The red arrow indicates the measured speed on the speedometer, and the yellow arrow is the actual speed as measured by some other device (a gps for example). We see that at 30 mph, we are actual traveling at 25 mph, there is a negative bias of 5 mph. At 50 mph.

We are actually traveling at 50 mph so there is no bias at this speed. At 70 mph however, we are actually traveling at 85mph! This is a positive bias of 15 mph. Our local police officer would be very interested in this result. In an ideal world, bias would not exist, but since we don't live in an ideal world, we know it does, and we would like the bias to be predictable. That leads us to the next measurement characteristic; Linearity.

Linearity is the measure of the bias over the range of interest in the measured samples. In our example below, we see that at 30 mph there is a negative bias of 5 mph and that as we proceed up the scale of measurement, the bias increases to plus 15 mph at 70 mph. This is NOT a linear response.

*The chart*

If you look at the chart below the speedometers below, you will see that the actual speed is not a linear line, its more quadratic (curved) than straight. This is useful information for us. First, it tells us that we can not apply a simple correction factor for bias across the range of measurement. If we were to apply a correction factor to the speed based on either end of the measurement range, results at the opposite end of the range will not be accurate. Secondly, the linearity tells us that the error grows as speed increases and that measurements at the high speed end of the range are more suspect and risky than measurements at the low speed end of the range.

Finally, lets talk a bit about sample selection. Hopefully through this discussion. You have seen that one of the most important aspects of setting up a Gage R&R Study is the choice of samples to measure.

*Linearity and Bias*

Remember the purpose of our study from earlier; determine if our measurement system can produce reliable results that can be used in decision making. Either for our six sigma project or in the actual measurement of quality. In order to know about things like Bias and Linearity of response, we must measure samples that span the range of interest of our measurement.

*What does this mean? *

Lets say for instance that using our speedometer example from above, that we have an upper specification of 65 mph and a lower specification of 30 mph. If we were to chose the measure at 50 mph because that result is in the middle of the range of interest, we get a very different picture of our capability to measure speed than if we measure at either end and outside the range of specification.

If we only measured 50 mph, we would incorrectly conclude that our speedometer is accurate and precise, with no bias. We would not be able to comment on linearity and this would result in our surprise at getting a speeding ticket for going about 12 mph over the limit at 65 mph.

If we measure across the range of interest we then can add linearity to our understanding and know that we should not be confident in results near the upper specification of 65 mph. This tells us that we should set our upper specification for the speedometer somewhere in the area of 58 mph (measured) to always be under 65 mph (actual).

*Conclusion of Gage R&R*

Gage R&R is a very useful tool in your six sigma tool box. It is also vital to ensuring that customer receive good product that meets their needs. Gage R&R studies are constructed to tell us how much confidence we can have in the measurement system. Gage studies can tell us where we need to improve the measurement system. Through analysis of the statistics that come along with the study.

We can determine if person variation is causing issues or if the device itself is the source of variation. In any case, the gauge study is a versatile tool to help identify improvement needs and improve quality.

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