2026/02/23 by Andrew T. Karl
Engineering · Computer Science · Decision Sciences · #Advanced Electrical Measurement Techniques #Direction-of-Arrival Estimation Techniques #Scientific Measurement and Uncertainty Evaluation
paper · pdf · doi:10.1007/s00362-026-01877-0
I--MR charts commonly estimate the process standard deviation σ via the span-2 average moving range divided by the unbiasing constant d2; unlike the unbiased sample standard deviation (S/c4), 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/d2, 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/c4 is almost entirely an adjacency effect.