Abstract:I--MR charts commonly estimate the process standard deviation $\sigma$ 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.
From: Andrew Karl [view email]
[v1]
Mon, 23 Feb 2026 16:15:07 UTC (19 KB)
[v2]
Sat, 7 Mar 2026 03:22:07 UTC (8 KB)
[v3]
Tue, 10 Mar 2026 11:21:17 UTC (8 KB)
[v4]
Sat, 18 Jul 2026 16:28:10 UTC (30 KB)
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