Calculation of the centerline for S-Charts

[glarson]

You are correct, A3 is just 3/(c4*sqrt). But the calculation of sbar is not simply the average of the standard deviations. Why can we just average standard deviations to get sbar? The hint is that the process is stable and the assumption is that each Si is equal....correct?

I feel like I'm in college again taking a final oral exam

"Why can we just average standard deviations to get sbar?" I mistakenly answered this question earlier, didn't I? When all of the n's are equal the pooled std dev formula reduces to the ave of the si results. Using preliminary data an unbiased estimate Sbar/c4 = sigma is used. the pooled std dev is replaced with sbar, and since n is the same for all subgroups the formula becomes ((sum si)/m)/c4 = sigma. (sum si)/m is the average of all the si since sbar reduces to the just the aveage of the all the si, it becomes the logical choice for the centerline of the s chart.

Right?

 

[Bev D]

Actually it has more to do with the sources of variation and what you are doing with the SDs.


We must use variances for combinations of standard deviations from different sources as standard deviations from different variation sources do not add linearly (the old right triangle theory)

In SPC when we are trying to determine the average within subgroup SD, the variation source is the same and the 'average' is calculated to remove the natural sampling variation from the different subgroups. We aren't combining different sources...

 

[Bev D]

Bev, thank you.

You have given me substantiation that I have been doing it properly. Unfortunately, several of the charts that I've been reviewing were not calculated properly, specifically those that involve variable subgroup sizes. With the help of your post (it's funny how it's easier for people to believe a stranger than a coworker) I hope correcting matters will go more smoothly than it has. Yes, I will be showing your responses to coworkers. Hope you don't mind.

Yeah - I have no idea why we believe complete strangers, but we do. Heck we even believe an unknown blogger over established experts...

When I came across the use of the third calculation for unequal variances in a chart, I had a hard time containing my bewilderment. I was told that this calc was used because each subgroup has its own std. dev. I explained as you did, that if each subgroup has a different population std dev. then the process is not in a state of control. Though they agreed with my statement, they did not think the formula was wrong. Since I'm not a statistician, of course I don't know more than an engineer...<sigh> Sorry for venting.

Again, Thank you.

The formula isn't wrong, it jsut doesn't apply to SPC. One of the fundamental precepts of SPC is that the control limits are calculated on a stable process. Each subgroup from a stable process will exhibit some variation due to simple sample error, that is why we average them using either the first or second formula.

If the sampel standard deviations are really different (the S chart will exhibit out-of-control conditions) then the process isn't stable.

 

[Barbara B]

In addition to the useful postings from other Cove-members you'll find attached an overview of the formulas for standard deviations provided earlier, in addition the unbiased versions with c4 (if available) and a small example in which the results of this formulas are compared.

A good repository for statistical questions is the Minitab knowledgebase (it's free, there are many formulas described and it's not necessary to have a Minitab licence at hand). One paper proposed there deals with Formulas for estimating the standard deviation for variables control charts.

Hope this clarifies a little bit the differences in practice,

Barbara

 

[Bev D]

It is probably worth remembering that the factors A3, B3 and B4 used to calculate the control limits for xbar, S charts include the unbiasing factors like c4 so if we stick with the basic formulas we are OK.