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Suppose you are monitoring a process by pulling samples

of the product at some regular interval and measuring one

critical quality characteristic (X). Obviously, you will not

always get the same result when measure for X. Why

not? There are many sources of variation in the process.However, these sources can be grouped into three

categories:

variation due to the process itself

variation due to sampling

variation due to the measurement system

These three components of variation are related by the following:

where t2is the total process variance; p

2is the process variance; s

2is the

sampling variance and ms2is the measurement system variance. Note that the

relationship is linear in terms of the variance (which is the square of the standard

deviation), not the standard deviation.

For our purposes here, we will ignore the variance due to sampling (or more

correctly, just include it as part of the process itself). However, for some processes,

sampling variation can greatly impact the results. Thus, we will consider the total

variance to be:

Remember geometry? The right triangle? The Pythagorean Theorem? The above

equation can be represented by the triangle below.

The total standard deviation, t, for a measurement is equal to the length of the

hypotenuse. The process standard deviation, p, is equal to the length of one side

of the triangle and the measurement system standard deviation, ms, is equal to the

length of the remaining side.

You can easily see from this triangle what happens as the variation in the product

and measurement system changes. If the product standard deviation is larger than

the measurement standard deviation, it will have the larger impact on the total

standard deviation. However, if the measurement standard deviation becomes too

large, it will begin to have the largest impact.

Thus, the objective of improving a measurement system is to minimize the %

variance due to the measurement system:

% Variance due to measurement system = 100(ms2/t

2)

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The gage R&R study focuses on ms2. In a gage R&R study, you can break down

ms2into its two components:

chart below shows the results by part by operator.

Interaction Sum of Square and Degrees of Freedom

We will make use of the equality stated earlier to find the interaction sum of

squares. This equality was:

SST= SS0+ SSP+ SS0*P+ SSE

SS0*P= SST- (SS0+ SSP+ SSE)

SS0*P= 32.317 - (1.63 + 28.909 + 1.712)

SS0*P= 0.065

The same equality holds for the degrees of freedom:

df0*P= dfT - (df0+ dfP+ dfE)

df0*P=44 - (2 + 4 + 30)

df0*P= 8

SummaryThis is the first of a multi-part series on using ANOVA to analyze a Gage R&R

study. It focused on providing a detailed explanation of how the calculations are

done for the sum of squares and degrees of freedom. We will finish out the ANOVA

table as well as complete the Gage R&R calculations in the coming issues.

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Measurement Systems Analysis (Gage R&R)

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September 2012

This month's newsletter is the second in a multi-part series on

using the ANOVA method for a Gage R&R study. This method simply uses

analysis of variance to analyze the results of a gage R&R study instead of the

classical average and range method. The two methods do not generate the same

results, but they will (in most cases) be similar. With the ANOVA method, we will

break down the variance into four components: parts, operators, interaction

between parts and operators and the repeatability error due to the measurement

system (or gage) itself.

Thefirst part of this seriesfocused on part of the ANOVA table. We took an in-

depth look at how the sum of squares and degrees of freedom were determined.

Many people do not understand how the calculations work and the information that

is contained in the sum of squares and the degrees of freedom. In this issue we will

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complete the ANOVA table and show how to determine the % of total variance that

is due to the measurement system (the % GRR).

In this issue:

The Data

The ANOVA Table for Gage R&R The ANOVA Table Results

Expected Mean Squares

The Variances of the Components

The % Gage R&R

Summary

Quick Links

As always, please feel free to leave a comment at the bottom newsletter.

The Data

We are using the data from ourDecember 2007 newsletteron the average and

range method for Gage R&R. This newsletter also explains how to set up a gage

R&R study. In this example, there were three operators who tested five parts three

times. A partial picture of the Gage R&R design is shown below.

Operator 1 tested each 5 parts three times. In the figure above, you can see that

Operator 1 has tested Part 1 three times. What are the sources of variation in

these three trials? It is the measurement equipment itself. The operator is the same

and the part is the same. The variation in these three trials is a measure of the

repeatability. It is also called the equipment variationin Gage R&R studies or

just with the within variationin ANOVA studies.

Operator 1 also runs Parts 2 through 5 three times each. The variation in thoseresults includes the variation due to the parts as well as the equipment variation.

Operator 2 and 3 also test the same 5 parts three times each. The variation in all

results includes the equipment variation, the part variation, the operator variation

and the interaction between operators and parts. An interaction can exist if the

operator and parts are not independent. The variation due to operators is called the

reproducibility. The data we are using are shown in the table below.

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Operator Part Results

A

1 3.29 3.41 3.64

2 2.44 2.32 2.42

3 4.34 4.17 4.27

4 3.47 3.5 3.645 2.2 2.08 2.16

B

1 3.08 3.25 3.07

2 2.53 1.78 2.32

3 4.19 3.94 4.34

4 3.01 4.03 3.2

5 2.44 1.8 1.72

C

1 3.04 2.89 2.85

2 1.62 1.87 2.04

3 3.88 4.09 3.674 3.14 3.2 3.11

5 1.54 1.93 1.55

The ANOVA Table for Gage R&R

In most cases, you will use computer software to do the calculations. Since this is a

relatively simple Gage R&R, we will show how the calculations are done. This

helps understand the process better. The software usually displays the results in

an ANOVA table. The basic ANOVA table is shown in the table below for the

following where k = number of operators, r = number of replications, and n=

number of parts.

The first column is the source of variability. Remember that a Gage R&R study is a

study of variation. There are five sources of variability in this ANOVA approach: the

operator, the part, the interaction between the operator and part, the equipment

and the total.

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The second column is the degrees of freedom associated with the source of

variation. The third column is the sum of squares. The calculations with these two

columns were covered in thefirst part of this series.

The fourth column is the mean square associated with the source of variation. The

mean square is the estimate of the variance for that source of variability (notnecessarily by itself) based on the amount of data we have (the degrees of

freedom). So, the mean square is the sum of squares divided by the degrees of

freedom. We wi l l use the mean squ are informat ion to est imate the var iance of

each source of var iat ion this is the key to analyzing the Gage R&R resul ts.

The fifth column is the F value. This is the statistic that is calculated to determine if

the source of variability is statistically significant. It is based on the ratio of two

variances (or mean squares in this case).

The ANOVA Table ResultsThe data was analyzed using the SPC for Excel software. The results for the

ANOVA table are shown below.

Source df SS MS F p Value

Operator 2 1.630 0.815 100.322 0.0000

Part 4 28.909 7.227 889.458 0.0000

Operator by Part 8 0.065 0.008 0.142 0.9964

Equipment 30 1.712 0.057

Total 44 32.317

Note that there is an additional column in this outputthe p values. This is the

column we want to examine first. If the p value is less than 0.05, it means that the

source of variation has a significant impact on the results. As you can see in the

table, the operator by part source is not significant. Its p value is 0.9964. Many

software packages contain an option to remove the interaction if the p value is

above a certain valuemost often 0.25. In that case, the interaction is rolled into

the equipment variation. We will keep it in the calculations herethough it has little

impact since its mean square is so small.

The next column we want to look at is the mean square column. This column is an

estimate of the variance due to the source of variation. So,

MSOperators= 0.815

MSParts= 7.227

MSOperators*Parts= 0.008

MSEquipment= 0.057

You might be tempted to assume, for example, that the variance due to the

operators is 0.815. However, this would be wrong. There are other sources of

variation present in all put one of these variances. We must use the Expected

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Mean Square to find out what other sources of variation are present. We will use 2

to denote a variance due to a single source.

Expected Mean Squares

As stated above, the mean square column contains a variance that is related to the

source of variation in the first column. To find the variance of each source of

variation, we have to use the expected mean square (EMS). The expected mean

square represents the variance that the mean square column is estimating.

There are algorithms that allow you to generate the expected mean squares. This

is beyond the scope of this newsletter. So, we will just present the expected mean

squares.

Lets start at the bottom with the equipment variation. This is really the within

variation (also called error). It is the repeatability portion of the Gage R&R study.

The expected mean square for equipment is the repeatability variance. The

repeatability variance is the mean square of the equipment from the ANOVA table.

Now consider the interaction expected mean square which is given by:

Note that the EMS for the interaction tern contains the repeatability variance as

well as the variance of the interaction between the operators and parts. This is

what is estimated by the mean square of the interaction. The parts expected meansquare is shown below.

Note that the EMS for parts contains the variances for repeatability, the interaction

and parts. This is what is estimated by the mean square for parts. And last, the

expected mean square for the operators is given by:

The EMS for operators contains the variances for repeatability, the interaction andoperators. This is what the mean square for operators is estimating.

The Variances of the Components

We can solve the above equations for each individual 2. Repeatability is already

related directly to the mean square for equipment so we dont need to do anything

there. The other three can be solved as follows:

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We can now do the calculations to estimate each of the variances.

Note that the value of the variance for the interaction between the operators and

parts is actually negative. If this happens, the variance is simply set to zero.

% Gage R&R

The Measurement Systems Analysis manual published by AIAG (www.aiag.org)provides the following definition: The measurement system variation for

repeatability and reproducibility (or GRR) is defined as the following:

GRR2=EV

2+ AV

2

where EV is the equipment variance and AV is the appraiser (or operator)

variance. Thus:

The total variance is the sum of the components:

We can use the total variance to determine the % contribution of each source to

the total variance. This is done by dividing the variance for each source by the total

variance. For example, the % variation due to GRR is given by:

The results for all the sources of variation are shown in the table below.

Source Variance% of

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TotalVariance

GRR 0.1109 12.14%

Equipment(Repeatability)

0.0571 6.25%

Operators(Reproducibility)

0.0538 5.89%

Interaction 0.0000 0.00%

Parts 0.8021 87.86%

Total 0.9130 100.00%

Based on this analysis, the measurement system is responsibility for 12.14% of the

total variance. This may or may not be acceptable depending on the process and

what your customer needs and wants. Note that this result is based on the total

variance. It is very important that the parts you use in the Gage R&R study

represent the range of values you will get from production.One of the major problems people have with Gage R&R studies is selecting

samples that do not truly reflect the range of production. If you have to do that, you

can begin to look at how the results compare to specifications. We will take a look

at that next month as we compare the ANOVA method to the Average and Range

method for analyzing a Gage R&R experiment. You could also use a variance

calculated directly from a month's worth of production in place of the total variance

in the analysis.

Summary

In this newsletter, we continued our exploration of the using ANOVA to analyze a

Gage R&R experiment. We completed the ANOVA table, presented the expected

mean squares and how to use those to estimate the variances of the components,

and showed how to determine the %GRR as a percent of the total variance.

In the next newsletter, we will compare the ANOVA method to the Average and

Range method for Gage R&R.

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