# Total Variation Formula - Can someone provide a source for this formula

#### MasterBB

##### Involved In Discussions
Total variation.
(Sigma)T = Sqrt [(Sigma square product varaition)+0.5*(Sigma Square Error)]
Can someone provide a source for this formula.

#### Marc

##### Hunkered Down for the Duration with a Mask on...
Staff member
Can anyone help with this? My Thanks, in advance!

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
MasterBB: it would help others research this with a few more details.

for example, the .5 multiplier of sigma_e comes form the sample size correction factor (n-1)/n. In the particular document - and it's direct reference/source - that you found this in, the number of repeat measurements of each part is 2. Since n=2, the correction factor = (2-1)/2 = 1/2 = .5

the direct source of this approach is Donald Wheeler's "Problems with Gauge R&R Studies". This article includes his references.

Is your question concerning
• the derivation of the correction factor
• the appropriate use of the correction factor for estimating the measurement error
• or is it why this approach would be used to estimate the total standard deviation in a gauge R&R study?

#### MasterBB

##### Involved In Discussions
Bev,

The formual that I am referring to is I beleive from a Document that was posted here (possibly by you)..
Title: Statistical Procedure for Measurement Systems Verification and validation.
Page 9.
Formula - Total Variation.
Also, the Interclass Correction Coeficient.. What is the source?

Are they both from Wheelers? Is there a specific book that cites this?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
The most direct references are included below.

Without knowing exactly what your questions are – other than for specific references – I will caution you that the formula in the attachment you are looking at for total standard deviation is for a very specific application within a Gauge R&R study where there are at least two repeat measured for each part and a limited sample size for the study. I rarely use this formula instead recommending the use of historical data to calculate the total standard deviation using the traditional formula. This approach alleviates the need to estimate a total standard deviation based on a small sample that specifically and intentionally ‘adds’ the measurement error to the observed results in the study. But more importantly it also avoids the larger impact of the bias that can result from samples that are representative of the range of the process variation but are not representative of the distribution of that process variation. This bias can result in substantial over or under estimation of the total standard deviation.

Again, the direct reference for using the sample correction factor multiplier for measurement error is:

Donald J Wheeler, Craig Award Paper, “Problems With Gauge R&R Studies”, 46th Annual Quality Congress, May 1992, Nashville TN, pp. 179-185.

This is repeated in the book:
Evaluating The Measurement Process, Second Edition, SPC Press, 1988 by Donald J Wheeler, Richard W Lyday.

For references on the ICC (in addition to those above):

Donald J Wheeler, “An Honest Gauge R&R Study”, Manuscript 189, January 2009. http://www.spcpress.com/pdf/DJW189.pdf

Donald S. Ermer and Robin Yang E-Hok, “Reliable data is an Important Commodity”, The Standard, ASQ Measurement Society Newsletter, Winter 1997, pp. 15-30.

Ronald Fisher, Statistical Methods for Research Workers

Futrell, David, “When Quality is a Matter of Taste, Use Reliability Indexes”, Quality Progress, Vol. 28, No. 5, May 1995, pp. 81-86

A google search will also help as Intraclass correlation coefficients are common statistics.
(be careful in your typing as interclass coefficients are different than intraclass…

#### MasterBB

##### Involved In Discussions
Bev,

Thanks for the reply..
What I am trying to do is to figure out the Sigma Repeatability(Measurement Error).....
At this point I don't have a Gage R&R; So, I am doing a Type 1 Gage Study / Analyzing historical data...

Would you say then, it's safe to report my calculated standard deviation as my measurement error from the historical data?

Note: It's also unfortunate that the data collected were all performed by the same operator.

My next step, is to analyze the data via GLM.

Once agaign, thanks.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
At this point I would ask what you think a type 1 study is and what kind of data do you have to work with?

was it one part measured many times by a single operator?
several parts measured multiple times by a single operator?
Many parts measured once by a single operator?

Useful answers come from detailed specific questions with data for us to look at.

#### Miner

##### Forum Moderator
Staff member
At this point I would ask what you think a type 1 study is and what kind of data do you have to work with?

was it one part measured many times by a single operator?
several parts measured multiple times by a single operator?
Many parts measured once by a single operator?
A Type 1 gauge study uses one operator on one reference part to measure bias and repeatability only.

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
yes Miner thanks.
however, from some of MaterBB's questions and statements I can't figure out what he is really asking or if he understands what a type 1 gage study means.

MasterBB can you please clarify what your question really is? the total standard deviation formula and ICC questions you have posted in this thread are not relevant to a type 1 gage study as assessed by Minitab.

Last edited:

#### MasterBB

##### Involved In Discussions
Bev & Miner,

Thanks,
1- I am using Type 1 Gage Study to evaluate the capability of my measurement system.
2- I am using historical data to calculate (Sigma Repeatability) by using the range formula/d2..

3- Next steps is to do a Gage RR.

4- I can't calculate Sigma Reproducibility as I only have one operatper's data.

5- So, with that said: GRR Calculation for Variance MS = Variance Repeatability + Variance Reprod. Thus, I am only calculating Sigma Repeatability per #2.

I do, have the standard deviation from the historical data;but one may still ask what is my Measurement system variation which I don't have at this point....
Is it possible to extract that number; my guess is no...

What do you think?

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