In Individuals and Moving Range (I-MR) charts, the process standard deviation is often estimated by the span-2 average moving range, scaled by the usual constant $d_2$. Unlike the sample standard deviation, this estimator depends on the observation order: permuting the values can change the average moving range. We make this dependence explicit by modeling the order as an independent uniformly random permutation. A direct application of the law of total variance then decomposes its variance into a component due to ordering and a component due to the realized values. Averaging over all permutations yields a simple order-invariant baseline for the moving-range estimator: the sample Gini mean difference divided by $d_2$. Simulations quantify the resulting fraction of variance attributable to ordering under i.i.d. Normal sampling, and two NIST examples illustrate a typical ordering and an ordering with strong serial structure relative to random permutations of the same values.

翻译：在个体与移动极差（I-MR）控制图中，过程标准差常通过跨度2的平均移动极差（经常数$d_2$缩放）进行估计。与样本标准差不同，该估计量依赖于观测顺序：对数值进行排列可能改变平均移动极差。我们将顺序建模为独立均匀随机排列，从而明确揭示这种依赖性。通过全方差定律的直接应用，将其方差分解为顺序引起的分量和实际数值引起的分量。对所有排列取平均可得到移动极差估计量的简单顺序不变基准：样本基尼平均差除以$d_2$。仿真量化了在独立同分布正态抽样下可归因于顺序的方差比例，并通过两个NIST示例分别说明典型顺序与具有强序列结构（相对于相同数值的随机排列）的顺序。