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No macros or special functions are required but it does take a while to set everything up. Many people using special software just plug in the numbers without understanding what is being calculated.

If you follow this article you will fully understand the maths. As more of these factors are varied it can be expected that the variation in measurement results will increase.

This leads to two extreme conditions of precision; repeatability and reproducibility. Repeatability is the minimum condition for precision in which the above factors are held constant while reproducibility is the maximum condition in which all of these factors vary. Often some intermediate measure of precision is relevant in which all of the possible factors effecting reproducibility are not varied since some of these will be maintained constant for the process under consideration.

Will different operators be involved, or measurements take place in different locations, with different environments? The differences between the reproducibility conditions should represent differences encountered in the process being studied.

By applying Analysis of Variance ANOVA it is then possible to determine the individual variance components due to the part variation, the repeatability of measurements and the reproducibility between different operators. The calculation of variance components and standard deviations using ANOVA is equivalent to calculating variance and standard deviation for a single variable but it enables multiple sources of variation to be individually quantified which are simultaneously influencing a single data set.

This can then be referred to while reading the article and once you have a full understanding of how it works it will be easy to adapt it to your own studies. For a real study at least 10 parts, 3 operators and 3 measurements is recommended. The example spreadsheet is divided into two tables to make things clearer. The first table, shown below, has a separate row for each measurement made in the study.

Columns A to D contain the inputs recorded for the measurements in the study. The subsequent columns in this table are then used to calculate means and squared differences for each measurement as explained below. The second table in the example spreadsheet, shown below, is used to calculate various values which summarize the complete data set. Intermediate calculations are used to finally calculate the variance components and standard deviations for the; variation between the actual parts; repeatability; the reproducibility due to different operators; and some other sources explained in the following sections.

The Grand Mean is first calculated Cell D24 which is simply the mean for all measurement values. The formula used to calculate the Mean for Part is repeated in column E of the spreadsheet on each row so the mean for the part is given for each measurement made, the formula in cell E3 looks like this:. The formula used to calculate the Mean for Each Factor Level which represents repeatability is slightly different to that used for the Part and the Operator. So for each measurement the mean of all measurements of the same part by the same operator is found.

Once the relevant means for each measurement value have been calculated the grand mean is subtracted from each one and the difference is squared. These values are given in columns H, I and J. The total sum of squared differences, given in column K, is simply the square of the difference between each individual measurement value and the grand mean.

The final stage in calculating the sums of the squared differences is simply to sum the values in the columns H, I, J and K, the resulting sums are given in Table 2 of the example spreadsheet in cells O3, P3, Q3 and R3 respectively. When following the spreadsheet method of calculation the n terms are not explicitly required since each squared difference is automatically repeated across the rows for the number of measurements meeting each condition.

The sum of the squared differences for part by operator interaction, given in cell S3, is simply the residual variation given by:. The numbers of different parts n Part , of operators n Op and of repetitions of the measurement of each part by each operator n Rep are given in cells O3, P3 and Q3 respectively. This is calculated in Excel by counting the number of unique number values in the column containing the ID numbers not counting blank cells or text values which, for the Part ID is given by.

These values are then used to calculate the degrees of freedom DF for each factor using the below equations and given in cells O5, P5, Q5, R5 and S5 of the example spreadsheet. It is then possible to calculate the mean squared difference for each factor by dividing the corresponding sum of the squared differences by the degrees of freedom. These values are given in O6, P6, Q6 and S6 on the example spreadsheet. At this stage the similarity with the calculation of a simple variance should be quite apparent.

The significance of the part by operator interaction on variation should then be determined by first calculating the F-statistic in cell S7 which is the Mean Squared value for Part by Operator interaction divided by the Mean Squared value for Repeatability.

If the interaction is significant then the above values of the mean squared differences are used to calculate components of variance. The alpha value to test against is given in cell O10 and an if statement is used in P10 to state whether the interaction is significant. This value is calculated in cell Q The variance components for each factor can now be calculated. In some cases the equation used depends on whether Part by Operator interaction is included in the model and in these cases an IF statement is used to check the value in P10 and select the correct equation accordingly.

When the part by operator interaction is not significant the variance component for part-to-part variation is calculated using. It is possible for these equations to return a negative value in which case the value should be set to zero, therefore the variance component for part by operator interaction is calculated in cell O21 using the formula:.

Since this is only included when this factor is significant and negative values are set to zero the Excel function is. When there is significant interaction it is given by. The standard deviations for each factor are simply the square root of the corresponding variance component.

I would not, however, recommend using this approach to analyze production data. There are too many steps leaving room for human error. Also, what is the calculation for Number of Distinct Categories? Thank you! I am having the same issue Jo had back in where I get a negative value in cell S4. This causes problems in cells O8:S8 and beyond.

Is there a particular reason for a negative value in S4 and is there a best method to deal with it? Using Excel I understand the importance of determining the contribution of the part, appraiser and measurement equipment variation as a portion of the total variation. Is it important to understand the contribution of all the sources of variation as a portion of the tolerance as well?

I assume this would give you an indication if the proposed test is capable of providing test results that can meet the tolerance. If this is important, can you propose a method for analyzing it? It was really a great help to me!

Please clarify how to calculate it. Also what the mean of Part operator interaction please describe with formula if any. Muelaner, Thank you for the detailed explanation. I am doing destructive testing. Thank you for the explanation. I have a question, What is the criteria to accept a measuring system?. That is for you to decide.

But these are just arbitrary limits. I will be publishing a method of setting limits optimally to minimise cost in the near future so watch out for that. I wanted please to ask, which cell in the spread sheet refers to the total process variation? Cell R23 is the total variaton exressed as a standard deviation including the actual part variation as well as the measurement variation.

Did I input a formula incorrectly or is that by design? Is there a way to get around that? Much obliged by your article, thanks. When I use my data, I get a negative value in cell S4. Can you email your spreadsheet with the error so I can check it? Also what version of Excel are you using?

How do I expand the worksheet to accommodate three appraisers with ten samples each, and not damage the formulae that exist within the worksheet now? Columns E to K, just copy the formulas down the columns. And that should work! Hi Dr. Muelaner, Should the repeat measurements be done same time or separate days or separate shifts? Regards, Bhanu. Generally they should be done at the same time, this is always the case. Repeatability is the variation between measurements under all of the same conditions.

But the real definition of reproducibility would include under changed conditions so that might mean different operator, different time, different temperature, different part, different insturment, different method etc.

You must consider what influences will vary for your measurement process. This is why an uncertainty evaluation approach is required. Why is. It appears to be manually entered. Please advise — thanks,. This spreadsheet is really intended to explain the calculations. An Alpha of 0. In your J column the sum does not do what you state ie it is not column G minus grand mean…?

How does this change if at all when the number of repeats is not the same for all parts? Say part 5 is measured 3 times by operator 1 but only 2 times by operator 2. Hi: Is there a specific reason why you are using an alpha of 0. Regards, -Deep. Maybe that will help me understand why the spreadsheet is formatted the way it is. Have you tried running the same figures through MiniTab?

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## ANOVA gauge R&R

You have done everything right. You carefully selected the parts to reflect the range of production. You carefully selected the operators to do the testing and randomized the run order for the parts. Each operator tested each part the required number of times.

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## Gage Repeatability and Reproducibility (Quantitative)

No macros or special functions are required but it does take a while to set everything up. Many people using special software just plug in the numbers without understanding what is being calculated. If you follow this article you will fully understand the maths. As more of these factors are varied it can be expected that the variation in measurement results will increase. This leads to two extreme conditions of precision; repeatability and reproducibility. Repeatability is the minimum condition for precision in which the above factors are held constant while reproducibility is the maximum condition in which all of these factors vary. Often some intermediate measure of precision is relevant in which all of the possible factors effecting reproducibility are not varied since some of these will be maintained constant for the process under consideration.

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## Gage Repeatability and Reproducibility, Gage R&R in Excel

A measurement systems contains variation from three main sources: the parts or products being measured, the operator aka appraiser taking the measurements and the equipment used to perform the measurements. The main focus of the study is on determining whether the measurement system is adequate for its intended use. Such a study evaluates several parts that are repeatedly measured called trials by multiple operators. A typical study is done by three operators who measure 10 parts at least three times each. The level of variation is then calculated among the operators and parts, across the trials. The operators should randomly sample the production across all the major sources of process variation parts produced, equipment used, equipment calibration, shift, time between measurements, etc.