# Measurement Error on pure Graphite Coating on Tubes - To Error or Not to Error

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Hi all, my company Silicon Carbide Coats pure graphite. we put parts in our furnace and coat the insides of long tubes. What we have been doing to determine the coating thickness is to first measure the ID before and after coating. Our inspector then substracts the diameters and records that value as our coating thickness. Our application engineers (bless their little hearts) have told one of our customers we can garrentee a minimum thickness of 0.0015" of coating. Our customer bit the bait and now asks up for our measurement error. We did a gage R&R on the bore mic. My question is if we are using two measured point values and doing a substraction to determine the coating thickness, our upper bound uncertainty limit would be the 0.0015" + 2(5.15 * total gage error standard deviation). I am thinking that using a multiplier of 2 for two measured points instead of 1. This new limit would be the coating thickness that if we were that or thicker we could garrentee the customer that we meet the specification of minimum coating thickness of 0.015". Any thoughts?

P.S., Hi Bev!

#### Michael_M

##### Quite Involved in Discussions
Re: To Error or Not to Error

Not sure I understand, but coating thickness is per side is it not. So if you measure the ID before coating (lets say 1.0000) and after coating (lets say .990) your coating thickness would be .005. It would not matter if you used a tri-micrometer or a bore gauge, it would always be divided by 2 as it is measured per side.

The area of concern that might crop up is the thickness differences across from each other (more coating on the 'north' side than the 'south' side), the calculation would give the average thickness not the actual.

##### Involved In Discussions
Re: To Error or Not to Error

Not sure I understand, but coating thickness is per side is it not. So if you measure the ID before coating (lets say 1.0000) and after coating (lets say .990) your coating thickness would be .005. It would not matter if you used a tri-micrometer or a bore gauge, it would always be divided by 2 as it is measured per side.

The area of concern that might crop up is the thickness differences across from each other (more coating on the 'north' side than the 'south' side), the calculation would give the average thickness not the actual.
Yes we do that. When I multiply the equation by 2 is (I think) for having two variable inputs (machined and coating) and their respective measurement uncertainty values. For example, machined diameter is 1.0000 +/-0.0002 (5.15 x total gage error Standard Deviation) and the coated diameter is 0.9900 +/-0.0002 (same gage same error). If I need to have our coater at a minimum diameter to ensure a calculated thickness does meet at say a 95% confidence level, then I need to add twice the gage error to the print tolerance of minimum 0.0015 inch coating thickness. Gage error forces us to be over that thickness by some thickness for us to make a garrentee of that. Hope this helps.

#### NumberCruncher

##### Inactive Registered Visitor
Re: To Error or Not to Error

Yes we do that. When I multiply the equation by 2 is (I think) for having two variable inputs (machined and coating) and their respective measurement uncertainty values. For example, machined diameter is 1.0000 +/-0.0002 (5.15 x total gage error Standard Deviation) and the coated diameter is 0.9900 +/-0.0002 (same gage same error). If I need to have our coater at a minimum diameter to ensure a calculated thickness does meet at say a 95% confidence level, then I need to add twice the gage error to the print tolerance of minimum 0.0015 inch coating thickness. Gage error forces us to be over that thickness by some thickness for us to make a garrentee of that. Hope this helps.

You need to add the variances, not the standard deviations.
Standard deviation of measurement 1 = S1
Variance of measurement 1 = S1^2 (S1 squared)

Standard deviation of measurement 2 = S2
Variance of measurement 2 = S2^2 (S2 squared)

If you add or subtract the measurements, the total (1 standard deviation) error in your result will be:

square root(S1^2 + S2^2)

This is the combined uncertainty of the calculation (measurement 2) - (measurement 1)

Also, you are using the wrong multiplier. The 5.15 value, I guess, is from the MSA manual. The above calculations give you the calculated thickness (let's call it T) and the total (1 standard deviation) error (let's call it S).

The estimate for the thickness is T+/- S.

Because this calculation uses 1 standard deviation, this would give you a 67% confidence interval. If you wanted about 99% as a confidence interval, you would use T +/-1.96S.

If you use T+/- 5.15S, I don't know what confidence interval this would give you, as I don't have tables that go out that far. As it is in the area of "six sigma", it would give confidence intervals in the area of 99.999?? %.

However it may well be wise to use a much bigger multiplier than 1.96 for safety. As Michael_M has said, you don't know how symmetrical the coating is and all you are accounting for is the gauge error.

NC

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Thanks Number Cruncher and Michael. Very useful information. Yes I did use the AIAG 5.15 statistic. I also use minitab and that is the multiplier we use for the Gage R&R to get our total gage R&R Study Variation in Minitab. I can see you would not use that great of a multiplier for the variances. We typically use a 95% confidence limit. I am courious Number Cruncher where you got the 1.96 value? Could you tell me what table that came from? Also one quick question, how would you set up a Gage R&R for a one sided tolerance ( Minimum Thickness of 0.0015" with no maximum thickness? Could I add my total uncertainty limit to 0.0015" at a 95% CL and then do a Gage R&R in minitab using continous data as attribute data and do an attribute Gage R&R in minitab? Could I still perform one using continous dats?
Thanks for the help.

#### NumberCruncher

##### Inactive Registered Visitor

The 1.96 comes from the standard Normal distribution. If you click on the link below,

http://www.zscorecalculator.com/index.php

you get to a live calculator. In the top box, click on 'Middle, (Equal area)'. You can then adjust the 'Left z score'. This adjusts the standard deviation (z score). If you set the z score to 1 you will see that the total area (red plus blue) is about 68%.

This is a 68% confidence interval for a standard Normal distribution. The box just above the curve shows the various statistics. The bottom box gives 'Percentile or probability'

If you now adjust the Left z score to 1.96, you will see that the total area under the curve is now 95%, or 95% confidence interval.

This all applies to a perfect Normal distribution. Real life distributions are rarely perfectly normal, but normality is a common assumption.

I can't give you much help with Minitab. I don't have a personal copy and I have only had access to it at work for a few weeks. I think that there is an option for a one sided tolerance, but I can't give you any detail on how to access it, especially over the weekend when I'm not at work!

NC

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Re: To Error or Not to Error

You need to add the variances, not the standard deviations.
Standard deviation of measurement 1 = S1
Variance of measurement 1 = S1^2 (S1 squared)

Standard deviation of measurement 2 = S2
Variance of measurement 2 = S2^2 (S2 squared)

If you add or subtract the measurements, the total (1 standard deviation) error in your result will be:

square root(S1^2 + S2^2)

This is the combined uncertainty of the calculation (measurement 2) - (measurement 1)

Also, you are using the wrong multiplier. The 5.15 value, I guess, is from the MSA manual. The above calculations give you the calculated thickness (let's call it T) and the total (1 standard deviation) error (let's call it S).

The estimate for the thickness is T+/- S.

Because this calculation uses 1 standard deviation, this would give you a 67% confidence interval. If you wanted about 99% as a confidence interval, you would use T +/-1.96S.

If you use T+/- 5.15S, I don't know what confidence interval this would give you, as I don't have tables that go out that far. As it is in the area of "six sigma", it would give confidence intervals in the area of 99.999?? %.

However it may well be wise to use a much bigger multiplier than 1.96 for safety. As Michael_M has said, you don't know how symmetrical the coating is and all you are accounting for is the gauge error.

NC
Thanks number cruncher. Maybe it's just my feeble attempt to understand this. Why would Ford use the multiplier of 5.15 as they state in their MSA third addition book? It states on page vi, "Historically by convention, a 99% spread has been used to represent the 'full' spread of measurement error, represented by a 5.15 multiplying factor (where
σGRR is multiplied by 5.15 to represent a total spread of 99%)" Am I completely missing the concept here? You recommend using the z score of 1.96 for 95% confidence level. Minitab defaults to 6 as the multiplier in its calculations. Use of 1.96 vs 5.15 or even 6 is vastly different. So now I am confused as to the two methods of calculating the error. Do you think I should just ignore Ford's calculation and strictly use the normal table in all error calculations? i.e., find the standard deviation and set the confidence level to 95 using the standard normal tables and then multiply the standard deviation by the corresponding z value to get my +/- error?
Appriciate you taking the time to explain this to me.
Regards,

#### Allattar

##### Inactive Registered Visitor
Re: To Error or Not to Error

Looks like there is some confusion about confidence intervals and variation.

Standard deviations are measuring the variation in individual results. If we were talking about a population the mean +/- 1 standard deviation would cover 68.4% of the normal distribution, +/- two standard deviations covers 95.4% of a normal distribution, +/- three is 99.7, and +/- 2.575 is 99%.

That is the individual results.

Confidence intervals refer to were a population parameter is estimated to be. For instance a mean of 10 with a 95% CI of +/- .5 would give a CI of 9.5 - 10.5. We could say we are 95% confident the population mean could be found within 9.5-10.5.
The confidence interval though is not simply +/- a number of standard deviations. In a t-test the confidence interval is based off the t-distribution and is found from the middle part of that t-distribution for that many degrees of freedom. (Very roughly it is about +/- two standard deviations/square root of sample size, but only if you have about 20 samples or more).

So when you say you want to be 95% confident that product meets a certain thickness. Do you mean 95% meet that spec, which actually is being 95% confident that 95% will meet that spec.
Or are you implying something else?

Also gage R&R is not capability, it is only the ability to measure, if you can measure then you can use a capability analysis to see if you meet spec.

Minitab will also handle one sided tolerances in gage R&R and capability. The mean is used to help calculate one sided tolerance in gage R&R, and you are comparing how much of the sample is in spec with capability.

Im a little bit confused with all the replies here, and I could have jumped on the wrong end of the stick, but it looks like you are using a gage to tell you if you are in spec.