# Unbiased Estimators for Sigma? Is 's' an alternative to RBar/d2?

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#### George Tirebiter

I would be interested to know if there is an alternative estimator for sigma than Rbar/d2 that can be used for sampling. I realize that S^2 is an unbiased estimator for sigma^2, but S is not unbiased for sigma. Is there any option other than Rbar/d2? I am writing a small program for HP calculators and would like it if the only input required was the string of Xbar values. Any help would be appreciated.

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D

#### Don Winton

In Grant and Leavenworth’s Statistical Quality Control, 6th Edition, page 107, there is an alternative to the Rbar/d2 unbiased estimate of sigma. Hope that helps.

Regards,

Don

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I was better but I got over it.

G

#### George Tirebiter

Thank you for the pointer, I am still a bit confused on whether or not S is a valid estimator for sigma or not, some texts seem to think so while others are adamant that it is not. Computationally, S is easier to calculate than Rbar/d2, for my application any way. So does anyone have any insight into the question as to S or not to S?

D

#### Don Winton

<font color=#df0016><BLOCKQUOTE>I am still a bit confused on whether or not S is a valid estimator for sigma or not, some texts seem to think so while others are adamant that it is not.</BLOCKQUOTE></font>

The answer to that question depends on one thing: How accurate do you want the estimate?

From my reference above, an unbiased estimate of sigma can be obtained by sigma=s/c4. From “Applied Statistical Methods”, Academic Press, Page 437, c4 varies by the sample size n. For sample sizes greater than 30, c4 is greater than 0.9914. For sample sizes greater than 60, c4 is greater than 0.9958. Again the question becomes: Is this error between s and sigma acceptable? If not, c4 increases as n increases.

Determine which value of c4 is acceptable, ensure n is greater than the corresponding value of c4 and use s rather than sigma.

Just a thought.

Regards,

Don

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I was better but I got over it.

G

#### George Tirebiter

Thank you for a clear and concise answer. Someday perhaps I will delve into the deep theoretical meaning of it all, but suffice to say that for now I will take a pinch of caution with my S.

D

#### Don Winton

<font color=#c001c0> <blockquote>Thank you for a clear and concise answer.</blockquote></font>

No problem. The “Applied Statistical Methods” I referenced above can also be found in various other texts (Juran, Feigenbaum, Grant and Leavenworth, etc.)

<font color=#c001c0> <blockquote>…for now I will take a pinch of caution with my S </blockquote></font>

For practical usage, I have found that when n is greater than 30, the difference between sigma and s is a minor thing and when n is greater than 60, the difference is negligible. Enjoy.

Regards,

Don

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I was better but I got over it.

G

#### George Tirebiter

Yes that is true as n increases S approaches sigma, my real question is that using rbar/d2 is fine when working with subgroups if you have a value for rbar. My problem was in writing this code for the HP 48G calculator that it isn't really easy to do without having people have to enter a great deal of data. I need to check my Juran's for those C4 values.

G

#### George Tirebiter

Well, it seems my curiosity is not quite dead. I was looking over some data and fitting some curves to some histograms, it seems that unless the actual data distribution is near or perfectly normal Rbar/d2 is much less accurate than S. SO what gives? Any ideas? I took some data sets with 25 subgroups of 5 points each and calculated the S for the subgroup averages and the overall mean of the subgroup averages. I basically calculated based on sub-group averages not on individuals. Then I calculated the Rbar/d2 and plotted the standardized histogram and normal curves using S and Rbar/d2 and found that every time S was a more accurate representation of the data distribution. Is Rbar/d2 a hold over from the pre-calculator days when it was easier to find Rbar than to calculate S? Any thoughts?

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