ISACA COBIT 5 – Measure (BOK V) Part 6

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- January 26, 2023

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**16. Gage R&R – Range Method (BOK V.C.1) **

So starting with the range method of gauge R and R study, we have already talked about this, that this provides a quick estimate of GRR, gauge, repeatability and reproducibility. And the problem here is this doesn’t split the variation into two pieces. This just gives you one single value. So this is basically a quick way to look at whether your measurement system is capable or not. So what we do here is we take two operators and we ask them to take the measurement of five parts one time each.

So we pick five pieces, we take two operators and we ask these two operators to take the measurement of these five pieces. So piece number one, measurement of operator number one is recorded and operator number two is recorded. Then for the piece number two, measurement of operator one and measurement of operator two. And same way we do for all these five pieces, whatever is measured by operator one and operator two, that is recorded. In ideal case, operator one and operator two should measure the same value for each part. So part one they should have the same value, part two they should have the same value. But that doesn’t happen if there is a variation.

And this variation could be because of the gauge, this variation could be because of the operator itself. And as we said earlier, in range method, we are not splitting that. We are not trying to find out whether this variation between these two operator ratings is because of operator or it is because of the gauge itself. So that we are not doing in range method. Let’s move on to the next slide and look at one simple example and what all calculations are done. Calculations here are very easy. So let’s look at that on the next slide. So here on this slide on the right side, if you see I have listed down five parts, part number 12345. And I have noted down whatever reading was taken by operator A or the appraiser A and appraiser B. So the first part, appraiser A notes down the reading of 100 mm.

Appraiser B looks at that and finds out the value as 10 one. There is a gap of 1 mm between these two readings. So that is recorded in the range column on the extreme right. Same thing with part number two. Appraisal one finds out the value of the measurement as 99. Appraisal B finds out that value to be 10 two. So there is a difference of 3 mm between these two readings and that we do for all these five parts. So once I have done this for these five parts, I take the range average of the range. The ranges are 1320 and two, the average of that comes out to be 1.

6 and that is known as R bar. From R bar you can get the estimate of the standard deviation by dividing a constant which is called d two. So let’s understand for now that the value of d two in this case will be 1. 19. And if I divide 1. 6 by 1. 19, that gives me a value of 1. 34. So that is basically the standard deviation because of measurement. 1. 34 sigma because of measurement. Now I have a process and in the process also there is some variation. Historically, if I look at my process and in historical records I see that my process is running with sigma value of or the standard deviation value of 2. 23. This is something which I know from my past operations. If I divide my GRR which is 1. 34 which is the standard deviation because of measurement divided by the process standard deviation, multiply that by 100, that will give me percent GRR. In this case, 100 multiplied by 1. 34 divided by 2. 23. So with this my percent GRR comes out to be zero six. What does that mean?

That means that 60% of variation is consumed by measurement only. That is too much. In ideal case this should be something around 10% 10% variation because of measurement error. That is something which is accepted. But once you have 60% of the variation which is coming as a part of measurement, that is not acceptable. So with this I can say that this measurement system is not capable. It’s not capable for this particular process. So this measurement system needs improvement. So this was my gauge R and R study using the range method. And you will see that this is very quick and easy. You can just pick five pieces, take two operators, ask them to take the measurement and you can do quick calculation to find out whether your measurement system is good or that needs improvement. Or maybe that might need more investigation. And if you think you need to do more investigation then you might want to go to the next level of method which is average and range method. Or you might want to go with the ANOVA method. So with this we complete the range method. Now let’s move on to the average end range method of cash.

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**17. Gage R&R – Average and Range Method (BOK V.C.1) **

To the second method of GRR gauge, repeatability and reproducibility studies using average and range method. Earlier we talked about the range method, which was a quick method. Here you would have a bit more calculations. Earlier we just got one number of GRR that in our example was 60%. And with that we said that this particular measurement system is not capable. Now here, once we go to average and range method, we will split that GRR into two parts, repeatability and reproducibility. How much of this variation is because of gauge and how much of this variation is because of the appraiser or the operator, that we will find out since the calculations are complex. So what I will do is I will take an example, do the calculation using Sigma Excel software and compare those results with the quick manual calculation we will be doing on the same axl sheets. And as we have already talked in average and range method, we don’t find out the interaction between the appraiser and the gauge. Interactions are not calculated here. Interactions are calculated in the next method which is ANOVA. So right now in this case, we will be just splitting the GRR into two parts. Variation because of the appraiser and variation because of the gauge. So here I have my Sigma excel. The next thing I will do is I will go to measurement system analysis. Click on that and click on template. Open gauge, R and RMSA template. Here, let me do that. So this is the template in this template. Now I need to put values in yellow area. So what does this need is? One thing this needs is the process tolerance. What is the process tolerance? The upper and the lower specification limit.

Then it’s looking at standard deviation multiplier. In earlier sessions also I have talked about this value being 5. 15 or six. At different places you might see this value changing from 5. 15 to six. Let’s keep it six here. And if I go down, this tells me that default value is six. But change this to 5. 15 if AIAG convention is being used and AIAG is related to automobile industry. So there you use 5. 15. Let’s not go into that detail here. Let this be six here. Now what I need to do is I need to take 90 measurements here, three operators, operator A, B and C, ten parts. I take ten parts from the production. These ten parts should represent the tolerance level, something near to the upper specification limit, something near to the lower specification limit, something in the middle. It should represent the broad range of the tolerance. Now, if I have selected those ten pieces, I give those ten pieces to three of these operators. And each of these operators takes three readings on each part.

So operator number one looks at the part number one. Three readings, reading number one, reading number two, reading number three. Same thing with the part two and same thing happens with the operator B and operator C. Once I put all these values here, this will give me whatever I want to calculate here. That’s all simple and plain. Let’s put some values here, hypothetical value, and understand how this thing works. So that I have already done on the next tab. Let’s look at that tab here, which is here. So here I have my lower specification limit as 90 mm, upper specification limit as 110, average value as 100. And those ten pieces I have given to these three operators, each taking three measurements and everything is recorded here. So operator number one, operator A looks at the part number one and takes three measurement which comes out to be 199, 99 takes part two, takes three reading, comes out to be 97, 100 and 198. And these are all hypothetical readings. I just made up these numbers. So once I’ve done that, this gives me this table. Now, let’s understand this. The first thing here is I need to understand three main things here.

One is total gauge R and R, which is coming out to be 58. 99, around 59%, which tells me that 59% of variation is because of the measurement itself. 59% of the overall measurement is because of the measurement system itself, which is too high. So that means our measurement system is not capable. Something like this we have done in the range method also where we got this GRR number, percent GRR number. But now what we do in average and range method is we get two more numbers. So let’s look at those two more numbers also which are here, repeatability and reproducibility. Reproducibility is coming out to be 14% and repeatability is coming out to be 57%. And you might see that these numbers do not add up. Like 57 plus 14 doesn’t make 58. Let’s not go into the detail, but squares of these will make the same thing. Like if I take a square of 00:57, square of 00:14, that will be basically the square of 00:58. Let me make it clear here that standard deviations don’t add up.

You don’t add up sigma, the standard division of the measurement and the standard division of the part. To give you the overall standard division, standard divisions are not added up. What we add up is the variances and variances are the square of that. You can add up the variance of the measurement and the variance of the part that will be the overall variance. What I need to find out is how did we reach at these numbers 57% as repeatability and 14% as reproducibility. So what I will do now hence forward is take another sheet and on that I do manual calculations. So if you are interested in that, keep on proceeding with this lecture. But if you think that’s getting too much, then you can skip this lecture. Let’s move on to the next tab where I’ve done all these calculations of finding out the repeatability and reproducibility. And which is here these are all the same numbers which we had same measurements for operator A, operator B and operator C.

Now, what calculations I have done here is first thing I have done is I have found out the average of these three measurements for each operator for each part. These three parts. The average is 99. 33. The range here is the difference between the highest and lowest so 100. -99 gives me one and if you see the formula here. Which tells the maximum of these three and minimum of these three, which is one same thing I have done for other 30 readings as well. 30 readings means ten parts and three readings each. Let’s say another example. Here an example of part one being measured by operator number B. The average of these three values is 99. 67. The range is three, which is 101. -98 so all these calculations have done once I’ve done these calculations which are highlighted here. The next thing I do is I find out the part average.

Now, part number one was measured 99. 33 by operator A, 99. 67 by operator B and 99 by operator C. So average of these three, this one, this one and this one comes out to be 99. 33. So on the average part number one is 99. 33. Same thing I do for ten other parts. So this is something which I’ve done as a part average. Let’s move forward here. What I’ve done is I have found out the X bar for a operator A, operator A. Whatever measurement this fellow has taken on these ten pieces. The average of that was 99. 33. Average of all ten pieces. Same thing with Operator B. Operator B had average as 100. 17. Operator C has an average of 99. 90. Now, you see that each operator has taken the equal number of measurement. These should be same. X bar A, x bar B, x bar C should be should have been same. But there is a difference. This is the difference which tells that there is operator to operator variation. These numbers will lead to operator to operator variation. Now, coming to R bar, r bar A is three point eleven, which is the average of these numbers, these ranges of operator A.

Now, these ranges basically, if you look at that part one was measured three times by the same operator. Actually it should have come same but why am I getting this difference is this difference is because of the measurement gauge. Because this is not because of the operator, because it’s the same operator. This is the measurement error because of the gauge variation, because of the gauge. This is represented here by R. First time, second time, third time. Average of that is R bar A, which is three point eleven. Same thing I do for Operator B. Three point eleven, operator C 3. 6. Now, what I do is I take R part. And R part is basically the range of these parts, part number one, part number two, part number ten, whatever is the range of that, that comes out to be our part which is 1. 78 x bar difference and x bar difference is basically the difference in x bar of these three people. X bar A, x bar B, x bar C. The difference of that is x bar difference. Whatever is the maximum out of that which comes out to be 100. 17 minimum is 99. 3. The difference between these two will be 1. 13. So this is x bar difference. X bar bar is the average of these 399. 3 100. 1799 point 90.

So that’s x bar bar and R bar is the average of R bar A, r bar B, r bar C. There are a little bit of formulas here. I will be attaching this sheet, you can download that, look into this. Take your time to understand. Now, coming to drawing the x bar chart and the R chart, we will be talking about these control charts later on as well once we go to the control phase of that. But let’s stick to the measurement system here. Upper control limit for the measurement is R bar bar multiplied by d four and lower control limit is r bar multiplied by d three. You will understand these constants in detail d once you go to control charts. So go to control charts. Once you complete that, you might want to come back here and relook at this lecture. So this gives me the upper and lower control limit for the range as 8. 4366 and upper control limit and lower control limit for x bar also as 96. 35 and 10 3. 0452. So with this if I draw my control charts so here I have my x bar chart and bottom I have the R chart. These numbers go here, 96. 36, 10 3. 4, that’s a lower and upper control limits and the bottom is the range chart. One thing which you need to remember here is if you, even if you don’t remember everything which I talked so far, what you need to remember here is all these points in the range should come within the upper and lower control limit which are coming here.

That’s fine. Just opposite to the control chart here in x bar chart, x bar chart, most of my points should be outside upper and lower control limit. This might not look very intuitive to you, this might look very fine that everything is in the control limit. This is fine. No, this is not fine here. When you are taking gauge R and R in gauge R and R, your 75% 80% points should be outside upper and lower control limit. Just remember that if you don’t remember anything else on this lecture, just remember that that these points should be outside which are not in this case. So that means there is a problem with the measurement system here. So coming to these values of equipment variation and appraiser variation. Equipment variation is repeatability. Repeatability is because of the gauge and reproducibility is because of the appraiser which is EV. Appraisal variation, these are given by these formulas. EV is equal to R bar multiplied by a constant k one. The k one which you can find it from here since we have three numbers of readings. So we take trials as three. Each operator is taking three readings. Same thing in case of reproducibility. Also there is a constant k two for which we take this value, because we have three appraisers here. And now if you put these in the formula where EV equipment variation is R bar multiplied by k one. And if you remember that repeatability is because of the gauge, repeatability is because of the gauge. And that’s why we are taking r bar bar.

If I go up and look at this R bar, and if you remember this R bar is coming because of one operator taking multiple readings. So this is because of the gauge issue. So R bar is related to gauge. And that’s what we are doing here is R bar bar multiplied by k one, which will give me the equipment variation and which is the repeatability. I can find out that repeatability value here. Tolerance Variation the tolerance if I divide into six sigma here by tolerance was anything between 90 to 110, that was my tolerance. Because my gauge variation is to be compared with the tolerance. If I have very wide tolerance, then I can accept gauge variation. Because if my tolerances are less tighter, then my gauge variation also needs to be tighter. And that’s what the whole measurement here is. So when I look at percent variation because of equipment, I need to divide that with the tolerance range and the tolerance.

Standard division or the TV here is 20 divided by 620 was the range 90 to 110 divided by six sigma, which will give me 3. 33, which was my TV. And if I divide my EV divided by TV, which gives me 57%. So my 57% is the problem because of equipment variation, something I need to do with my equipment, because my equipment is not behaving right and that I could even see looking at these readings. Also these readings should have been consistent. There is too much of variation here between these three readings of each operator looking at each part, that’s what is the problem here, which is coming out to be 60% now, looking at reproducibility, which is variation because of the appraiser here, if you look at this x bar difference multiplied by k two. And if I look at this x bar difference, x bar difference is the difference between these x bars. These averages should have been same once in a while. If one operator reads high, one operator reads low, but on the average they should be reading equal. But since there is a difference between the x bar here, which gave me this x bar difference and which is here being used for finding out the reproducibility and AV by putting these values in this formula is zero point 47 6771. And if I divide this with the TV, which is the tolerance, this gives me 14%. So 14% of variation is because of the appraiser and the gauge R and R will be the square of these two.

If I take square of EV and square of AV and add them and take a square root of that, that will give me GRR which comes out to be 59. 63%. This is similar to what was calculated automatically by Sigma XL. There might have been slight difference in the number. That is because of somewhere I would have rounded off these values. And what is the acceptable limit of that? Less than 10% GRR is acceptable. Between ten to 30% is marginally acceptable in certain cases where the criticality is low. But more than 30% GRR is not acceptable because that means your measurement system is inadequate. It. So this completes our discussion on finding out GRR using average and range method. Now, you can see that things are getting complicated. If I go for anoa method calculation, that’s going to be too lengthy. So what I will be using is I’ll be doing Sigma XL calculation and we will be just looking at the results of that and interpreting the results of GRR.

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**18. Gage R&R – ANOVA Method (BOK V.C.1) **

About three methods in gauge R and R study, range, average and range. And third method was ANOVA we have already talked about the range average and range. And now we are talking about the ANOVA method of doing gauge R and R. So, what I’m doing here is I have taken a sample from Sigma XL software. So, this file is a part of attachment in Sigma Excel here. If you look at this file, I have parts part one here, then part two. And then I have up to part ten. Then again part one, part two, part three, up to part ten and then again part one, part two, part three. These are checked by operator A. So the part one is checked by operator A three times and the value is recorded here. This is exactly what we did in average and range method. Only thing here is that numbers or measurements are different. There we had a measurement which was around 100. Here it’s 00:29. 41. So these are different numbers, it’s a different example. But approach is same three operators, operator ABC, ten parts part one, two, three, up to part ten. And each part checked three times by each operator. That’s what we have here. And three multiplied by three multiplied by ten is 90.

So we have 90 readings here. Now, what we want to do is using Sigma Excel, we want to analyze this, we want to do crossed gauge RnR study. And when I say crossed gauge R and R, we will be talking later about another thing which is called as nested gauge R and R, crossed gauge R and R, which is this case is done when you can take number of measurements on one piece. You can measure part number one three times by operator A, three times by operator B and three times by operator C. But if this was a destructive test, you cannot have that thing done. So for this, you need separate pieces each time. So you will need nine pieces instead of one piece on which three operators do three measurements coming back to the cross gauge R and R. So this is our data here. Now, what I need to do is I need to go to measurement system analysis and analyze gauge R and R crossed. So I just need to click on this. So before that I click on my data here. Go click on analyze gauge R and R crossed.

And I say that select entire data, move on. And here I need to put parts. Parts are in column part operator in column named operator and measurements in column named measurements. And as I earlier told, there are two ways to use Sigma Multiplier. We can use 5. 15 or we can use six. Let’s keep it six here. Confidence interval of 90%. And with that I want to display x bar R chart, multiverry, chart, everything I want to display. And with this I press OK, sigma XL will do the calculation and then we can look at the result of that. We are not going into the details of each calculation, how each calculation is done and what multiplication was done. We are not going into that. We will straight away go into the results of that. Try to understand what does that mean.

So now, Sigma XL has done all the calculations. So these are the results of that. In this result, let’s look at a few important items. One thing is let’s say standard division, multiplier, that’s six, which we selected as six. Another thing which we want to look at here is these three things which says that there were ten parts. We know that three operators, operator ABC and three replication. Each operator checked each part three times. So that’s fine. Now, here are two ANOVA tables. We are not going into details of that. One thing which you need to look at in ANOVA table is the p value. P value can tell you the whole story. And once you look at design of experiments, hypothesis testing later, once you learn that, you will understand that when p is low, low means less than zero five, then null must go. That means null hypothesis gets rejected. And I’m not going into details of that. But in plain simple thing, that means that these are key contributory factors here. So part and operators are key contributory factors. Now let me move to another sheet which I did earlier.

And I have my notes there. So let’s open that. So this is exactly the same thing, the same analysis. But here I have some additional nodes here which can help you in understanding these. This file will be attached. So you can download this file and look at these numbers and these nodes. So we talked about these p values. P is low, null must go. That means part and operator are key contributory factors. Looking at gauge R and R here, this gives me the total gauge R and R value of 27% of tolerance TV tolerance variation. 27% of tolerance variation is because of gauge, repeatability and reproducibility. And this is high. 27% is quite high. So if you look at these notes here, less than 10%, if you get that total gauge R and R, that is acceptable. Between ten to 30%, these are acceptable depending on your application, depending on the criticality. In certain places these might be acceptable, certain places these might not be. But if this value is greater than 30%, then your measurement system is inadequate. So right now what we have here is 27%.

So which is on quite high side. So our measurement system is not really good here. Now, this 27% also constitutes of two main parts. One is repeatability and the second is reproducibility. Reproducibility is 20. 9 and repeatability is 18. 42. These are also quite high numbers. So gauge itself gives a lot of variation. The operator also gives a lot of variation. Together these contributes to overall gauge R and R of 27. 86%. Another thing which you need to look here is variance percent, contribution of variance component. This one here, I’m getting gauge R and R as 7. 76, I’m getting reproducibility as 4. 37 and repeatability as 3. 39. And here when you look at these numbers, the guideline here is that less than 1% is acceptable. Here we have 7% which is on quite high side and between one to 9% is marginally acceptable depending on the situation. And that’s where our readings are falling into marginally acceptable under certain conditions where criticality is low.

So this is another thing which you need to look into. Another thing which you need to look here is NDC number of distinct categories. I have a separate lecture on that which you will see once you complete this. NDC is the buckets in how many buckets this data will be falling into. We need to have NDC number high. Here what I’m getting is NDC of 4. 9, which is let’s say roughly five. Once you look at that lecture, that will tell you what does that mean. But for now, what we need to understand is when NDC is five or more, that means your measurement system is acceptable. Three. If the NDC is three, then your measurement system can divide your readings, your measurements into three groups low, medium, high. Let’s not talk about that because there is a separate lecture on that. Look at that, that will make clear what NDC means. But for now, NDC of roughly five is marginally acceptable. So that’s all we need to look into these numbers when we do measurement analysis using ANOVA now coming to x bar R charts. Also this draws x bar R chart as well.

And if you remember when we talk about average and range method in average which was x bar chart, I said that in x bar chart most of your points should be outside the control limit and which was not in our example, in our example, most of the points in x bar charts were within the control limit. That was not a good sign, that was a different data set. This one is a different data set. Let’s see what happens in this data. How does points fall, whether these are within control limit or outside control limit, that also is drawn here by Sigma XL. So I click here and this gives me the control chart. So the top one is the x bar chart and the bottom one is the R chart. In R chart we expect everything to fall within these two control limits, the upper control and lower control limit.

The point should be falling between that which more or less is happening here, except this one point, which is part number four, operator B. This has given a range of 1. 2, which is more than the upper control limit. There might have something happened in this particular set of three readings which operator B took on part number four. So you might want to go back, look at that data and see if something unexpected happened during that measurement. So that was our chart. Now, if you look at X bar chart, in X bar chart, you see there are 30 dots here. In 30 dots, if you see there are eight dots which are within lower and upper control limit. In ideal case, 75% to 80% of these points should be falling outside control limits. Outside upper and lower control limits. Not within, only let’s say 2020 5% points should be falling within upper and lower control limit, which in this case is marginally true.

Like out of 30 points, if I take one photo of that, which should be seven, and here we have eight points which are falling, maybe we can say that this rule is more or less being followed here. That only 20% to 25% points falling within upper and lower control limits. So we are fine. Here the previous example which we took in that most of our points were within these upper and lower control limits. So that was not a good sign of a measurement system. So here, this particular point is fine. So this is what all you need to look at in ANOVA.

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