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.

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The two methods do not generate the same results, but they will in most cases be similar. In particular it focuses on the sum of squares and degrees of freedom. 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.

This means you have to have variation in the results. And, in looking at the results, I discover that each result is the same - for each part and for each operator. There is no variation. I am asked - Isn't it good that there is no variation in the results? It means that the measurement process cannot tell the difference between the samples.

This newsletter also includes how to analyze the results using the average and range method. 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:. 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. 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. Repeatability is the ability of the measurement system to repeat the same measurements on the same sample under the same conditions. It represents an assessment of the ability to get the same measurement result each time.

Reproducibility is the ability of measurement system to return consistent measurements while varying the measurement conditions different operators, different parts, etc. It represents an assessment of the ability to reproduce the measurement of other operators. In this example, there were three operators who tested five parts three times. Operator 1 will test 5 parts three times each. 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. Operator 1 also runs Parts 2 through 5 three times each. The variation in those results 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.

The variation in all results is the reproducibility. The operator is listed in first column and the part numbers in the second column. The next three columns contain the results of the three trials for that operator and part number. For example, the three trial results for Operator A and Part 1 are 3.

We will now take a look at the ANOVA table, which is used as a starting point for analyzing the results. In most cases, you will use computer software to do the calculations. This helps understand the process better. The first column is the source of variability.

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. The second column is the degrees of freedom associated with the source of variation. The degrees of freedom are simply the number of values of a statistic that are free to vary. For example, suppose you have a sample that contains n observations.

We use the sample to estimate something - usually an average. When we want to estimate something, it costs us one degree of freedom. So, if we have n observations and want to estimate the average, then we have n - 1 degrees of freedom left. The third column is the sum of squares SS associated with the source of variation. The sum of squares is a measure of variation.

It measures the squared deviations around an average. Remember what the equation for the variance is? The variance of a set of number is given by:. The sum of squares for the source of variation is very similar to the numerator. You just take the sum of squares around different averages depending on the source of variation. 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 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. Note the similarity to the formula for the variance above.

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 the ratio of two variances or mean squares in this case. The data above were analyzed using the SPC for Excel software. The total sum of squares is the squared deviation of each individual result from the overall average - the average of all results. The overall average of the 45 results is:. This equation is simply a fancy way of saying that you subtract the average from an individual result and square that result. This is shown in the figure below for the squared deviation of the first result.

There were a total of 45 results. We calculated the overall average for these results. This can also be calculated as nkr - 1. As mentioned before, you obtain the sum of squares by determining the squared deviations between two numbers. With the operator source of variability, you will obtain the squared deviations between the operator average and the overall average.

Algebraically, this is given by:. The table below shows how the calculations are done:. So, you can see that the sum of squares due to the operators is based on how the operator averages deviate from the overall average. There are three operator averages. Since we calculated the overall average, we lost one degree of freedom. The variability chart below shows the results by operator by part.

The horizontal blue line is the average for the operator. The horizontal green line is the overall average. The difference between those two lines is the deivation. The sum of square due to the parts is done in the same manner as for the operators except the average you are focusing on are the part averages. Algebraically, the equation for SS P is:. The table below shows the calculations. The original data has been sorted by part.

Again, you can see how the sum of square due to parts is based on how the part averages deviate from the overall average. There are five parts. Again, we calculated the overall average, so one degree of freedom is lost. The equipment sum of squares uses the deviation of the three trials for a given part and a given operator from the average for that part and operator. This can be expressed as:. Again, note that the sum of squares is examining variation around an average.

For the within variation, it is the variation in the three trials around the average of those three trials.

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## Gage Repeatability and Reproducibility (Quantitative)

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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## ANOVA gauge R&R

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.