Dear Experts,

I am confused by one question concerning the SPC control limit calculation.
I use Xbar-R chart as an example, UCL=Xbar+A2*R, and normally we also say the control limit UCL/LCL is Xbar+/-3sigma.

But actually they are not same, then my question is what's their relationship?

Thanks in advanced.
Steven

A2 = 3 / ( d2 * sqrt[n] )
n: subgroup size

A2 = 3 / ( d2 * sqrt[n] )
n: subgroup size

and sigma = R / ( d2 * sqrt[n] )

so:
UCL=Xbar+A2*R >> UCL=Xbar+3* (R / ( d2 * sqrt[n] )) >> UCL=Xbar+3 sigma

Great to know the information. And then I have another question. Why do we calculate the UCL by Xbar+A2*R instead of Xbar+3 sigma directly?

We can't just use the standard deviation because we don't know sigma!

Suppose you chart 1 point every hour. If you measured all the parts from that hour, then you would know the true standard deviation for that hour. Since that is ususally too expensive, only a sample is drawn. The challenge is to estimate what the true sigma is based on a small sample.

One good estimate is to use the range of the sample:
sigma = R / ( d2 * sqrt[n] )

Another approach uses the sample standard deviation. I don't have my books handy, but the equation is similar. As I understand it, the standard deviation approach gives a better estimate of the true standard deviation (espicially for big samples), but the range approach is much easier to calculate.

Tim F

Just to clarify something....sigma=Rbar/d2. d2 is tabled by n, so the sqrt of n is not necessary.

We can't just use the standard deviation because we don't know sigma!

Suppose you chart 1 point every hour. If you measured all the parts from that hour, then you would know the true standard deviation for that hour. Since that is ususally too expensive, only a sample is drawn. The challenge is to estimate what the true sigma is based on a small sample.

One good estimate is to use the range of the sample:
sigma = R / ( d2 * sqrt[n] )

Another approach uses the sample standard deviation. I don't have my books handy, but the equation is similar. As I understand it, the standard deviation approach gives a better estimate of the true standard deviation (espicially for big samples), but the range approach is much easier to calculate.

Tim F

Looks like I should have checked the books before posting.:o

Tim F

Thanks a lot for everyone. I got some idea concenring this point based on the discussion.

Hey Tim.

A2 = 3/(d2*sqrt(n))

CL = Mean +/- A2 * R

Example: n=12, d2= 3.258 and A2=0.266
in Excel =3/3.258/SQRT(12) >> 0.265815041

d2 include sqrt(n)????, that's funny. :bonk:

d2 DOESN'T

Hey Tim.

A2 = 3/(d2*sqrt(n))

CL = Mean +/- A2 * R

Example: n=12, d2= 3.258 and A2=0.266
in Excel =3/3.258/SQRT(12) >> 0.265815041

d2 include sqrt(n)????, that's funny. :bonk:

d2 DOESN'T
All I said was sigma is Rbar/d2 not Rbar/d2*sqrt(n)

All I said was sigma is Rbar/d2 not Rbar/d2*sqrt(n)

For individuals (moving range or IX chart ) >> sigma (X)= Rbar/d2

but for averages >> sigma(X_bar) = sigma (X) /sqrt(n)

It depends on the point of view, for sigma(X) your point is right, but for Sigma(X_bar) not.

Great to know the information. And then I have another question. Why do we calculate the UCL by Xbar+A2*R instead of Xbar+3 sigma directly?

The reason we do not calculate the Control limits directly from the standard deviation of the individual averages is that we would be circumventing the fundamental principles of how the control chart works (for subgrouped data).

A stable process (that is sampled with rational subgroups) will a population standard deviation equivalent to the within subgroup variation / square root of the sample size. (Between subgroup variation will be be small in comparison to the within subgroup variation). If the process MEAN changes, the sample average will then move outside of the the limits. (If the process standard deviation chagnes, we see it in the R chart or S chart, of course).

If we use the standard deviation of the individual averages we do not test the within sample variation to the between sample variation assumption/requirement...we will not see out of control conditions unless they are VERY large. By definition, the individual values of a roughly Normal distribution will all (99.7%) fall within +/- 3 sigma of the mean and sample averages tend to approximate a Normal distribution - according to the Central Limit Theorem...

For individuals (moving range or IX chart ) >> sigma (X)= Rbar/d2

but for averages >> sigma(X_bar) = sigma (X) /sqrt(n)

It depends on the point of view, for sigma(X) your point is right, but for Sigma(X_bar) not.
We might be dealing with using different references, but my table for d2 has the sqrt of n already there. For example, my reference (Duncan) has a d2 value of 3.08 for n=10. Therefore sigma would be Rbar/3.08. I also looked in other references and they all refer to Rbar/d2.

For individuals (moving range or IX chart ) >> sigma (X)= Rbar/d2

but for averages >> sigma(X_bar) = sigma (X) /sqrt(n)

It depends on the point of view, for sigma(X) your point is right, but for Sigma(X_bar) not.

Great, this is the traditional (Central Limit Theorem). That means sample mean is equal to population mean and Sigma (sample)=Sigma(population)/sqrt(n).

Great, this is the traditional (Central Limit Theorem). That means sample mean is equal to population mean and Sigma (sample)=Sigma(population)/sqrt(n).
:agree1: Good description, short and to the point.

I must add that Control Limits use Sigma (sample) or Sigma(X__bar) for "SPC for averages".

I hope and think that this journey has been enlighting for you.

Great, this is the traditional (Central Limit Theorem). That means sample mean is equal to population mean and Sigma (sample)=Sigma(population)/sqrt(n).
actually just to clarify your terminology a bit, the within sample standard deviation is not = the population standard deviation divided bythe square root of n. The standard deviation of the sample AVERAGES is = population standard deviation divided teh square root of n...the within sample standard deviation divided by c4 or the range divided by d2 ~ population standard deviation...

actually just to clarify your terminology a bit, the within sample standard deviation is not = the population standard deviation divided bythe square root of n. The standard deviation of the sample AVERAGES is = population standard deviation divided teh square root of n...the within sample standard deviation divided by c4 or the range divided by d2 ~ population standard deviation...
This brings up the whole issue of population standard deviation (sigma) and sample standard deviation (s). R/d2 is an estimate of the population standard deviation, not the sample standard deviaiton.

I am not convinced, but give up.

Could you try to obtain A2 from d2, and check the outcome? :truce:

I am not convinced, but give up.

Could you try to obtain A2 from d2, and check the outcome? :truce:
Let me try to clarify the confusion with a little more verbage. Xbar+/-A2*Rbar is like saying Xbar +/- 3*sample standard deviation. Because for controls charts the sample size is known. When we talk about estimating the population standard deviation (unknown), we just take Rbar/d2. If we wanted an estimate of the sample standard deviation, we would use Rbar/d2*sqrt(n).

A2 is 3/d2*sqrt(n) so you are correct that Rbar/d2*sqrt(n) does give the standard deviation, but I believe this is an estimate of the sample standard deviation or short term sigma.

A2 is 3/d2*sqrt(n) so you are correct that Rbar/d2*sqrt(n) does give the standard deviation, but I believe this is an estimate of the sample standard deviation or short term sigma.

Agree. :yes:

This brings up the whole issue of population standard deviation (sigma) and sample standard deviation (s). R/d2 is an estimate of the population standard deviation, not the sample standard deviaiton.
exactly.

A2 is 3/d2*sqrt(n) so you are correct that Rbar/d2*sqrt(n) does give the standard deviation, but I believe this is an estimate of the sample standard deviation or short term sigma.
uhm not exactly...
A2*Rbar is the estimate of the standard deviation of the SAMPLE AVERAGES or "between sample" std dev.
the "short term" standard deviation is an estimate of the WITHIN SAMPLE standard deviation of individual values. The infidels who invented capability indices, coined the phrase to describe the variation of individual values (or parts) without the 'between sample' variation (which is A2*Rbar)

It is confusing...especially when we cannot use formulas and diagrams except in attachments...

Let me see if I can summarize this: (without getting into Sigma-hat and such:( )

If you are plotting XBar values (subgroup averages), you need to compute 3 Sigma limits for subgroup averages.
Sigma_of_XBar = Sigma_of_individuals / sqrt[n]
Sigma_of_XBar = (RBar/d2)/ sqrt[n]

Now the limits are XBarBar +/- (3* Sigma_of_XBar)
= XBarBar +/- 3* (RBar/d2)/ sqrt[n]
= XBarBar +/- A2* RBar
and A2 = 3/( d2 * sqrt[n])

See:
http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm
http://www.isixsigma.com/forum/showmessage.asp?messageID=25669


Also:
The limits calculated without dividing by sqrt[n], ie. XBarBar +/- (3*RBar/d2) are called the Natural Process Limits (UNPL and LNPL)

: Ref: Understanding SPC- Don Wheeler

We might be dealing with using different references, but my table for d2 has the sqrt of n already there. For example, my reference (Duncan) has a d2 value of 3.08 for n=10. Therefore sigma would be Rbar/3.08. I also looked in other references and they all refer to Rbar/d2.Right, but I don't understand what the confusion is :confused:. My copy of Quality Control & Industrial Statistics - AJ Duncan ( an old old copy) lists the following values in Table M (Factors useful in the Construction of Control Charts):

for n=10, d2=3.078 and A2=0.308.

same as all other tables for control chart constants I have.

Right, but I don't understand what the confusion is :confused:. My copy of Quality Control & Industrial Statistics - AJ Duncan ( an old old copy) lists the following values in Table M (Factors useful in the Construction of Control Charts):

for n=10, d2=3.078 and A2=0.308.

same as all other tables for control chart constants I have.
I think we are in agreement...the issue is the estimate of sigma. Is it Rbar/d2 or Rbar/d2*sqrt(n)

uhm not exactly...
A2*Rbar is the estimate of the standard deviation of the SAMPLE AVERAGES or "between sample" std dev.
the "short term" standard deviation is an estimate of the WITHIN SAMPLE standard deviation of individual values. The infidels who invented capability indices, coined the phrase to describe the variation of individual values (or parts) without the 'between sample' variation (which is A2*Rbar)

It is confusing...especially when we cannot use formulas and diagrams except in attachments...

Within Sample Variation, not Between...

Between:
1- Obtain the standard deviation for the means SD(X_bar)
2- Correct the value with the c4 factor (SD(X_bar)/c4) to obtain an unbiased estimator for stdev for the averages
3- Multiply the estimate by sqrt(sample size) to obtain the "unbiased estimate for the stadard deviation for the individuals SD(X).

NOT A2* R_average (another way to skin the animal, but with different results) :mg:

I think I'll take a stab at this topic. I would like to start by thinking closer to the root of the issue - why some of these numbers are useful. By starting closer to the beginning, much of the following is review for many, but I realizd I have learned a fair amount about the topic by wading through all this myself, so hopefully some of you can gain from my struggles.

One additional challenge is simply notation - trying to name all the different ideas and trying to type all the different symbols.

Control charts look for stability in two areas - does the overall value stay about the same, and does the level of variation stay the same. For these we need to estimate several parameters about the process.


The approach is to draw a sample of n parts - say 4-10 parts - and measure some property of the parts.
X-bar is the average of these measurements.
R is the range of these measurements.
S is the standard deviation of these measurements.

Several sets of samples are drawn (usually at least 30) at different times
X-bar-bar is the average of the set of X-bar's
R-bar is the average of the set of R's
S-bar is the average of the set of S's
S(X-bar) is the standard deviation calculated from the set of X-bar's

If all the parts could be measured
µ would be the true mean
σ would be the true population standard deviation

ASSUMING the process is in control (all the variation is random - both short term and long term), then statistically it turns out that the variation in the individual parts is
σ ≈ R-bar/d2

The variation in X-Bar is
σ√n ≈ S(X-bar) ≈ R-bar/d2√n

The control limits for X-bar are three time this:
± 3σ√n ≈ ± 3S(X-bar) ≈ ± 3R-bar/d2/√n ≈ ± A3 x S-bar


If the process is not in control then the various estimates for the standard deviation, and hence for the control limits, will not be the same.
± 3 σ√n ≈ ± 3 S(X-bar)
and
± 3 R-bar/d2/√n ≈ ± A3 x S-bar
but the first pair are bigger than the second pair.

The second pair represent how well the process could do if all long term variation could be eliminated (the "within" variation). Since this is the ideal, these numbers are used to determine the control limits.

If there is additional variation between the samples above and beyond what is occurring within the samples, then the process is not in control and X-bar will show more variation than expected and the control chart will show a large number of points out of control.

I worked up a spread sheet with some sample random numbers to convince myself that this works - doing the calculations and making some control charts. I would be happy to share it if there is interest, but I fear this is long enough already.


Tim F

....
I worked up a spread sheet with some sample random numbers to convince myself that this works - doing the calculations and making some control charts. I would be happy to share it if there is interest, but I fear this is long enough already.
Tim, that would be interesting. Can you please post the spreadsheet here? Thanks.