I--MR charts commonly estimate the process standard deviation $σ$ via the span-2 average moving range divided by the unbiasing constant $d_2$; unlike the unbiased sample standard deviation ($S/c_4$), this estimator depends on ordering through adjacency, so permuting a fixed sample changes it. We formalize this by introducing an independent uniformly random permutation and applying the law of total variance, yielding an exact decomposition into a values component (variance of the permutation mean) and an adjacency component (expected conditional variance over permutations). The permutation mean is order-invariant and equals $\GMD/d_2$, where $\GMD$ is the sample Gini mean difference. Under i.i.d.\ Normal sampling, both components admit closed forms; the adjacency fraction converges to $0.3813$, and the familiar asymptotic efficiency loss relative to $S/c_4$ is almost entirely an adjacency effect.

翻译：I-MR控制图通常通过跨度2的平均移动极差除以无偏常数$d_2$来估计过程标准差$σ$；与无偏样本标准差($S/c_4$)不同，该估计量通过相邻性依赖于数据顺序，因此对固定样本进行排列会改变其估计值。我们通过引入独立均匀随机排列并应用全方差定律对此进行形式化，将其精确分解为数值分量（排列均值的方差）和相邻性分量（跨排列的条件方差期望）。排列均值具有顺序不变性，且等于$\GMD/d_2$，其中$\GMD$为样本基尼平均差。在独立同分布正态抽样条件下，两个分量均存在闭式解；相邻性分量占比收敛于$0.3813$，而相对于$S/c_4$常见的渐近效率损失几乎完全由相邻性效应导致。