This article generalizes the historical $S^2$ chart (of equal sample sizes) to subgroups of differing sample sizes. As a sequel to a previous article, Section 2 compares all estimators of process standard deviation for more than $M$ $>$ 1 subgroups of equal sample sizes from a normal universe. Section 3 provides the general probability control limits (CONLs) for the $S^2$ chart of unequal sample sizes. Section 4 uses the chi-square distribution to obtain the exact $\alpha$-level CONLs for the S chart. Section 5 proves that the average run length (ARL) of $S$ and $S^2$ charts are identical and tabulates the ARL for the most common sample sizes $n$ = 2, 3, $\dots$, and 12. Further, as expected, the ARL of $S$ charts always perform better than those of the corresponding $R$ charts.

中文翻译：

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使用 S2 或 S 控制图对不同样本量及其精确平均运行长度的子组进行质量控制

这篇文章概括了历史 ${秒}^2$图表（相同的样本量）到不同样本量的子组。作为上一篇文章的续篇，第 2 节比较了所有过程标准偏差的估计量超过$米$ $>$来自正常宇宙的 1 个样本大小相同的子组。第 3 节提供了一般概率控制限 (CONL)${秒}^2$不等样本量的图表。第 4 节使用卡方分布来获得准确的$\alpha$S 控制图的级别 CONL 。第 5 节证明了平均游程长度 (ARL)$秒$ 和 ${秒}^2$ 图表是相同的，并列出了最常见样本量的 ARL $n$ = 2, 3, $\dots$, 和 12. 此外，正如预期的那样，ARL 为 $秒$ 图表总是比相应的图表表现更好 $\mathrm{电阻}$ 图表。