Xbar-R Chart better than Xbar-S Chart when the subgroup size is small - Why?

in AIAG handbook,it says that if the subgroup size is small ,the Xbar-R chart is better than Xbar-S, can anybody help to explain this?

Any help explaining my results is appreciated.

Re: Xbar-R chart and Xbar-S chart

PeterWang said:

in AIAG handbook,it says that if the subgroup size is small ,the Xbar-R chart is better than Xbar-S, can anybody help to explain this?

Any help explaining my results is appreciated.

Click to expand...

Welcome to the forum

Check these: Xbar and R chart, Xbar and s chart

IMHO, Range variation estimates are less inflated by small samples.

S uses square in it's formula, so is prone to make it huge if an outlier is found.

Refer to this earlier thread on Xbar–R or Xbar-s.

Also: https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm

Atul Khandekar said:

Also: https://www.itl.nist.gov/div898/handbook/pmc/section3/pmc321.htm

Click to expand...

To quote the above article:

"However, s, the sample standard deviation is not an unbiased estimator of . If the underlying distribution is normal, then s actually estimates c4 , where c4 is a constant that depends on the sample size n."

and

"There is a statistical relationship (Patnaik, 1946) between the mean range for data from a normal distribution and , the standard deviation of that distribution. This relationship depends only on the sample size, n. The mean of R is d2 , where the value of d2 is also a function of n. An estimator of is therefore R /d2."


The other point I enjoyed:

"When do we recalculate control limits?

Since a control chart "compares" the current performance of the process characteristic to the past performance of this characteristic, changing the control limits frequently would negate any usefulness.

So, only change your control limits if you have a valid, compelling reason for doing so. "

My emphasis above. Nice article!

Ranges are easier to calculate manually.

While there are several debates (some theoretically valid, others mere gas) about the accuracy of the dispersion statistic based on sample size (degrees of freedom) and the underlying distribution, the reality is that Control Charts are not and never were intended to provide accurate and precise point estimates of the process performance. Any innaccuracy induced by the use of the range or the standard deviation (and both are innacurate) is overcome by the use of the properly calculated limits and the time series nature of the data layout (which has more to do with probability than with distributional statistics).

in short: it doesn't matter. use one of them. Shewharts' book was entitled "the economic control of quality", not the "exactly precise accurate determination of exactly what is going on with infinite confidence in the results"

Thank you for your help everyone.So, the conclusion is that S is better than R to detect the process variation,but for small sample size, not much is lost by using the R to instead of S,am i right?
Thanks a lot.

by the way the Wikis are incorrect: a cursory look reveals the following errors:

• Xbar R and Xbar S charts are NOT dependent on an underlying Normal distribution. They ARE dependent on a homogenous process stream but may have ANY shape.
• They are designed to 'catch' drifts of more than 1.5sigma

beware of Wikis. any yahoo can publish out there...and remember that just because it's published doesn't mean it's true.

PeterWang said:

Thank you for your help everyone.So, the conclusion is that S is better than R to detect the process variation,but for small sample size, not much is lost by using the R to instead of S,am i right?
Thanks a lot.

Click to expand...

yes. that is a correct answer

Bev D said:

by the way the Wikis are incorrect: a cursory look reveals the following errors:

• Xbar R and Xbar S charts are NOT dependent on an underlying Normal distribution. They ARE dependent on a homogeneous process stream but may have ANY shape.
• They are designed to 'catch' drifts of more than 1.5sigma

Click to expand...

Interestingly, the article only made a connection between the control chart constants and the normal distribution. In practice, they state:

"In the U.S., whether X is normally distributed or not, it is an acceptable practice to base the control limits upon a multiple of the standard deviation. Usually this multiple is 3 and thus the limits are called 3-sigma limits. This term is used whether the standard deviation is the universe or population parameter, or some estimate thereof, or simply a "standard value" for control chart purposes. It should be inferred from the context what standard deviation is involved. (Note that in the U.K., statisticians generally prefer to adhere to probability limits.)"

Nonetheless, Xbar R and Xbar S charts and the associated Western Electric rules are designed to detect variation from the mean. The control limits are a function of the mean, and the rules are concerned with activity about the mean. If that is appropriate action for the controlling the underlying distribution, then it is a great tool for any such processes. I would not say that is appropriate for any distribution, but it is good for many centralized distributions. There are the processes that the mean has no real value in understanding variation.

And that goes back to "Shewharts' book was entitled "the economic control of quality", and there is no specific tie from economic control and variation from the process mean. One tool he describes is appropriate for certain processes, but it is not the only tool in the tool box.