# Different Standard Deviations used in Control Charting

#### stm08007

##### Starting to get Involved
Hello,

I feel I am growing more and more comfortable with control charting, but I am trying to better understand why there are seemingly so many different standard deviations.

If I had 50-5 piece samples and wanted to do an X-bar and R chart, why should I use sigma hat for my standard deviation estimator? Why can't I just use the calculation for sample standard deviation? Same with the R-bar chart--- why can't I use the sample standard deviation of the range measurements-- why must I calculate sigma_R hat?

Hopefully this question makes sense-- I just want a clear explanation of why the "basic" standard deviation formulas (population and sample) we learn in high school would be inappropriate?

One other question... my book talks about "standard deviation of the sample average"--- I see others talk about "standard error of the mean"-- are these terms interchangeable?

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#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Hi stm08007, These are excellent and typical questions that many people who are new to control charting have. You are wise to ask them as unfortunately, many 'popular' references and instructors don't adequately address these issues.

The choice of which 'standard deviation' or more accurately the measure of variation is critical to the success of any control chart. get this wrong and your chart won't work right.

Essentially the control limits must be calculated using the within subgroup variation and not some other calculation of a standard deviation. This within sample subgroup variation is translated to the expected limits of variation of the subgroup averages (and the within subgroup variation itself) through standard formulas. if you are using a Xbar, R chart (typical for small sample sizes; historically used before statistical calculators and software as the Range was easy to calculate by operators on the line) the limits of expected variation come from the average range of the subgroups. If you are using a Xbar, S chart you use the average within subgroup standard deviation. If you are using the I, MR chart, the 'within subgroup variation' is the average moving range between sequential values.

WHY can't you use a different calculation like you learned in high school? The following explanation has a fair amount of detail

The foundation of how a control chart works is based on the homogeneity of the process stream. A homogenous process is one which the standard deviation and the average are controlled by the same factors. A non-homogenous process has a mean that is controlled by one or more factors that are different from the factors that control the standard deviation of the process. Control charts detect non homogeneity...non-homogeneity can result from a change in one of the factors or it can be a from a constant source in the process.

IF we have a homogenous process, the within subgroup variation will relate directly to the variation in the subgroup averages by this formula:
SD(Xbar) = Average SD(Within Subgroup) / (square root of n)

IF we do not have a homogenous process (in other words if an assignable cause is present) then the limits calculated using the above formula will either be way too tight or way to wide. This is how control charts work.

If you use a 'global' SD or the standard deviation of all of the individual values you will get limits that are too wide as this approach includes the within subgroup variation and the between subgroup variation (aka the SD of the subgroup averages).

Here are some articles that correctly address this very question. I urge you to read these and then come back with any further questions you may have:

Wheeler, Donald, “The Secret Foundation of Statistical Analysis”, Quality Digest, December 2015

Wheeler, Donald, “The Right and Wrong Ways of Computing Limits”, Quality Digest, January 2010

Wheeler, Donald, “Good Limits from Bad Data I”, Quality Digest, March 1997

Wheeler, Donald, “Good Limits from Bad Data II”, Quality Digest, April 1997

And yes the phrases "standard deviation of the averages" and Standard error of the mean" are the same thing. The latter is simply the 'formal' statistical name of the statistic while the former is the informal 'description' of the statistic

#### stm08007

##### Starting to get Involved
Thanks for the thorough response (and completely agree that it seems this is not adequately covered in any of the texts I've read).. I'll take some time to go through the articles (I actually already posted a question on here last week about one of your referenced articles), but I did already start with the first one.

In the last "Food For Thought" section- the author explains that an App in the App Store had 100 5 star ratings, 100 1 star ratings, 40 2 and 4 star ratings, and like 20 3 star ratings... the overall average was ~3 stars, but who does this represent--- this was to demonstrate the concept of homogeneity.

I'm a little confused as to what that is showing-- the author made a point of saying a normal distribution is not a requirement (clearly that is non-normal data), but homogeneity is---- isn't it possible that the true population is adequately represented by the sample used in the app ratings? If we were to discount a distribution like that, what besides normal data for that 1 to 5 star scale would be considered acceptable?

Hope my question makes sense- I was following the article pretty well until that last paragraph

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Which article are you referring to?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
OK you are referring to the "The Secret Foundation of Statistical Analysis”.

The data are clearly non homogenous as represented by the many who experienced the app as a 5 and those many who experienced it as a 1. not only is this not normally distributed, it is not random and clearly not homogenous. the causal system for the 5s and the 1s are VERY different. So the average clearly doesn't represent the total experience. Another way of thinking about homogeneity is that it's like 'homogenized milk'. if milk isn't homogenized the milk fat rises to the top and the lower layer is milk with very little fat. homogenizing mixes the milk so that it is a uniform distribution. Like a good shuffle of a deck of cards. In the case of the app ratings the distribution of ratings is then clearly not homogenous. Then this occurs the 'standard' statistics of the mean and the standard deviation may be mathematically correct but provide no real insight to the population. There are clearly TWO causal mechanisms at play.

The author is NOT saying that the sample doesn't represent the population. In fact he is saying that the sample DOES represent the population and the mean the and standard deviation do not adequately represent the population. He is not saying that we should discount this distribution, he is saying we cannot just blindly apply ‘normal distribution based statistics’ to this distribution. It is not a question of ‘acceptability’ of the distribution. It is a question of how best to analyze and understand the distribution. In order to do this we have to understand the fundamental role of homogeneity in our choice of analytical approaches.

Does this help?

#### stm08007

##### Starting to get Involved
Yes that makes sense. So if this instead looked like an exponential function (also not normal), it would satisfy the Homogeneity requirement (since it has ONE causal mechanism)?

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
So if this instead looked like an exponential function (also not normal), it would satisfy the Homogeneity requirement (since it has ONE causal mechanism)?
Not really. While in the example that Dr. Wheeler used, the non-homogeneity was clear from the histogram, it is not always the case. In fact, it usually isn't.
the only real way to determine if a process is non-homogenous is to plot the data in time series. I've even seen processes that have ~Normal distributions that are completely non-homogenous.

A control chart will certainly detect non-homogeneity (that's it's job).

I prefer to use a multi-vari first so that I can see all of the components of variation in a single chart...but that's a topic for a different time and thread

#### stm08007

##### Starting to get Involved
hmmm, that's pretty much what I'm driving at-- the author of the article seems to say normality is not a requirement but homogeneity is, which contrasts with the fact that it seems that you can't have homogeneity without normality?
Can you give me an example of a homogenous process that is not normal? I think that is the key to me understanding this!

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
…the author of the article seems to say normality is not a requirement but homogeneity is, which contrasts with the fact that it seems that you can't have homogeneity without normality?
The thing to understand is that the homogeneity – or lack of it – and the ‘best fit’ distributional model are completely unrelated to each other. I have seen processes that are clearly not normal, yet homogenous and approximately normal distributions that were clearly not homogenous. What is important is understanding the true nature of your process stream and what analytical method are necessary to analyze the process correctly.

Can you give me an example of a homogenous process that is not normal? I think that is the key to me understanding this!
Homogenous processes that are not normal: cycle time, paint thickness, fill volumes, etc. basically anything that has a natural or physically forced limit can be homogenous and will often not be ~Normal.
However, the really important thing to understand is that many processes are simply not Homogenous.

Probability Models do not Generate Your Data”, Donald Wheeler, Quality Digest, March, 2009

Wheeler, Donald, “Why We Keep Having Hundred Year Floods”, Quality Digest, June 2013

we are pretty far away from your original question so you might think of starting a separate thread to discuss homogeneity as opposed to standard deviation calculations.

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