This article discusses the (precision and accuracy) relative efficiencies of all estimators of a normal population standard deviation σ = σ_X for M > 1 subgroups of different sample sizes. We have examined the statistical properties of seven estimators of σ_X which are (1) the pooled estimator S_p, (2) the pooled unbiased estimator ${\widehat{\sigma }}_{\mathit{pub}}$, (3) the maximum likelihood estimator ${\widehat{\sigma }}_{\mathit{mle}}$, (4) Irving W. Burr's weighted ranges, (5 and 6) the last two Burr estimators, and the author's optimal estimator ${\widehat{\sigma }}_{\mathit{XO}}$. Unlike the case of M = 1 subgroup, our introduction section shows that the ${\widehat{\sigma }}_{\mathit{mle}}$ for M > 1 subgroups is the least accurate of σ estimators and should be avoided in all applications only when M > 1. As a consequence of our findings, modifications of at least two choices to control limits of S chart are provided. Part 1 of this article (Sections 1–6) compares estimators assuming a process is under statistical control, and Part 2 (Sections 7–10) discusses all statistical aspects of the S chart of varying sample sizes.

中文翻译：

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关于M> 1子组的sigma估计量和不同样本量的质量控制S图

本文讨论了（精度和准确度）正常群体标准差的所有估计的相对效率σ = σ _X为中号> 1子组不同的样品尺寸。我们已研究的7所估计的统计特性σ _X，其是（1）汇集的估计小号_p，（2）汇集无偏估计${\widehat{\sigma }}_{\mathit{酒馆}}$，（3）最大似然估计 ${\widehat{\sigma }}_{\mathit{le}}$，（4）Irving W. Burr的加权范围，（5和6）最后两个Burr估计量，以及作者的最优估计量 ${\widehat{\sigma }}_{\mathit{XO}}$。与M = 1子组的情况不同，我们的简介部分显示${\widehat{\sigma }}_{\mathit{le}}$对于M > 1个子组，σ估计值的准确性最低，只有在M > 1的情况下才应避免在所有应用中使用。由于我们的发现，提供了至少两个选择的修改以控制S图表的限制。本文的第1部分（第1-6部分）比较假定过程受统计控制的估计量，而第2部分（第7-10部分）讨论了不同样本量的S图表的所有统计方面。