Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition
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 adjacen…
Authors: Andrew T. Karl
Order Dep endence in the Mo ving-Range Sigma Estimator: A T otal-V ariance Decomp osition Andrew T. Karl ∗ Karl Statistical Services LLC F ebruary 24, 2026 Abstract In Individuals and Moving Range (I-MR) c harts, the pro cess standard deviation is often estimated by the span-2 a v erage moving range, s caled by the usual constan t d 2 . Unlik e the sample standard deviation, this estimator dep ends on the observ ation order: p erm uting the v alues can change the a verage mo ving range. W e make this dep endence explicit by modeling the order as an indep endent uniformly random p ermutation. A direct application of the law of total v ariance then decomposes its v ariance into a component due to ordering and a component due to the realized v alues. Av eraging ov er all p erm utations yields a simple order-in v ariant baseline for the moving-range estimator: the sample G ini mean difference divided by d 2 . Sim ulations quan tify the resulting fraction of v ariance attributable to ordering under i.i.d. Normal sampling, and t wo NIST examples illustrate a t ypical ordering and an ordering with strong serial structure relativ e to random p ermutations of the same v alues. Keyw ords: Individuals c hart; p ermutation distribution; Gini mean difference. 1 Bac kground and motiv ation Individuals and Mo ving Range (I-MR) charts commonly estimate the pro cess standard deviation using the span-2 a v erage moving range, MR(2), scaled by the usual constant d 2 ( Mon tgomery , 2019 ). The estimator is simple and widely used, but it is statistically less efficient than SD-based estimators under i.i.d. sampling ( Ho el , 1946 ; Cryer and Ryan , 1990 ; Braun and P ark , 2008 ). One reason is o verlap: each observ ation app ears in up to tw o moving ranges, inducing dep endence among the successiv e absolute differences. A second, conceptually distinct feature is order dep endence. F or a fixed set of observed v alues, the a verage moving range can v ary across p ermutations. This matters when observ ations hav e no inherent order or when the effective order is ambiguous ( Poots and W oo dco ck , 2012 ). In the con ven tional time-ordered I-MR setting, dep endence on order is usually inten tional: the moving range targets lo cal v ariation. In that setting, our p erm utation framing compares the observ ed adjacen t differences to those obtained from random reorderings of the same realized v alues, providing a compact diagnostic for serial structure. When order is arbitrary , the same framew ork instead regards ordering as a nuisance and quantifies how m uch v ariabilit y in the MR(2) estimator can arise purely from the chosen ordering, with the scaled Gini mean difference pro viding an order-inv ariant ∗ Corresp onding author. Email: akarl@asu.edu . 1 companion summary . Gini’s mean difference has also b een used directly as a disp ersion measure in qualit y-control charting (e.g., the G -c hart of Riaz and Saghirr ( 2007 ); Aslam et al. ( 2022 )). Related practice compares short-term sigma estimates to order-inv ariant summaries such as the o v erall SD (“short-term” versus “long-term” v ariation), and the v ariance-ratio/ stability-index literature pro vides additional to ols for that purp ose ( Cruthis and Rigdon , 1992 ; Ramirez and Runger , 2006 ). This pap er fo cuses on a probabilistic reframing that makes the order dep endence explicit. The main con tributions are: (i) a law of total v ariance partition of the v ariabilit y of the mo ving-range estimator under a random-order construct, whic h under an i.i.d. reference mo del for the observ ations equals the sampling v ariance of the usual fixed-order MR(2) estimator and therefore yields v alues and ordering comp onents; (ii) an order-inv ariant baseline obtained by av eraging ov er all p ermutations; and (iii) a compact p ermutation diagnostic for whether the observed ordering lo oks typical relative to random orderings of the same realized v alues. The remainder of the pap er is organized as follows. Section 2 defines the moving-range functional under a random-order mo del. Section 3 giv es the law of total v ariance decomp osition and shows that the p ermutation mean equals a simple Gini-based baseline. Section 3.3 records closed-form Normal- reference v ariances used for comparison. Section 4 presents a short simulation study quan tifying the order fraction under i.i.d. Normal sampling. Section 5 illustrates the p ermutation diagnostic on t w o NIST examples. Section 6 summarizes a practical workflo w for in terpreting the decomp osition in I-MR settings. 2 Setup and the mo ving-range functional Let X = ( X 1 , . . . , X n ) b e i.i.d. observ ations from a distribution with marginal standard deviation σ . Let Π b e a uniformly random p ermutation of { 1 , . . . , n } , indep enden t of X . Define the (unscaled) a verage moving range along order Π by MR( X , Π) := 1 n − 1 n X t =2 X Π( t ) − X Π( t − 1) . Define the scaled moving-range functional T ( X , Π) = MR( X , Π) d 2 . (1) F or an observed sequence with ordering Π obs (e.g., time order), w e write the observed-order estimate T obs := T ( X , Π obs ). Even when the observ ed order is meaningful, we introduce Π only as a mathematical device to define a p erm utation reference distribution obtained by randomly reordering the observed v alues. Under the i.i.d. reference mo del for X , this device also lets us in terpret V ar { T ( X , Π) } and its comp onents as sampling-v ariance quantities for the usual fixed-order estimator T ( X , Π obs ). W e do not require that the actual observed order Π obs w as generated by a random p ermutation. W rite E Π {· | X } and V ar Π {· | X } for exp ectation and v ariance with resp ect to the uniformly random p ermutation Π, holding the realized X fixed. Define the p ermutation mean ¯ T ( X ) := E Π { T ( X , Π) | X } . W e refer to ¯ T ( X ) as the p erm utation mean to distinguish it from the sample av erage moving range MR( X , Π). 2 The v ariance identities in this pap er are exact and do not require Normalit y . Normalit y en ters only through the conv entional choice of the scaling constan t d 2 (for i.i.d. Normal s ampling, d 2 = E | X 1 − X 2 | /σ = 2 / √ π ≈ 1 . 128 ( Montgomery , 2019 )). With this conv en tional Normal-reference constan t, E { T ( X , Π) } = σ under i.i.d. Normal sampling (for an y fixed ordering), whereas under other i.i.d. distributions T is generally not calibrated to σ unless d 2 is replaced by the appropriate distribution-sp ecific constant. This calibration relies on the usual serial-indep endence assumption b ehind I–MR charts: with serial dep endence, E { T obs } is determined b y the lag-1 joint distribution (through E | X t − X t − 1 | ) and generally differs from the marginal σ ev en when V ar( X t ) = σ 2 . 3 A la w of total v ariance decomp osition Under this framing, T ( X , Π) is random for tw o reasons: the realized v alues X and, conditional on those v alues, the ordering Π; the la w of total v ariance separates these contributions exactly . 3.1 V ariance split in to v alues and ordering comp onents Prop osition 1 (La w of total v ariance decomp osition for MR(2)). Let ( X , Π) and T ( X , Π) b e as in Section 2 . Then a direct application of the law of total v ariance yields V ar T ( X , Π) = E V ar T ( X , Π) | X + V ar E T ( X , Π) | X . (2) Under the i.i.d. reference mo del for X , the joint distribution of ( X 1 , . . . , X n ) is in v ariant to reordering. Therefore, for any fixed observ ed ordering Π obs , T ( X , Π obs ) has the same sampling distribution (and hence the same v ariance) as T ( X , Π) when Π is uniformly random and indep endent of X . Consequen tly , under i.i.d. sampling, ( 2 ) is a v ariance decomp osition for the conv entional MR(2) estimator. The term V ar { ¯ T ( X ) } (i.e., V ar { E ( T ( X , Π) | X ) } ) is the v alues comp onen t, and the term E V ar T ( X , Π) | X is the ordering comp onent. F or a fixed realized sample X = x , the within- sample ordering v ariability is V ar Π T ( x, Π) , which can b e estimated by Mon te Carlo p ermutations of x . A con v enient summary is the or der fr action , OrderF raction( n ) = E V ar T ( X , Π) | X V ar T ( X , Π) , (3) the prop ortion of the total v ariance V ar T ( X , Π) attributable to ordering under the i.i.d. reference mo del. Although w e write OrderF raction ( n ), the ratio in ( 3 ) generally dep ends on the sampling distri- bution of X (and on the scaling constant d 2 ); Section 3.3 fo cuses on the i.i.d. Normal reference, where it dep ends only on n . An equiv alen t form (using ( 2 )) is OrderF raction( n ) = 1 − V ar { ¯ T ( X ) } V ar { T ( X , Π) } . (4) 3.2 P ermutation mean equals a Gini baseline Prop osition 2 (Perm utation mean and GMD). F or an y fixed realization x = ( x 1 , . . . , x n ), let ¯ T ( x ) := E Π { T ( x, Π) } . ¯ T ( x ) = GMD( x ) d 2 , (5) 3 where GMD ( x ) = 2 n ( n − 1) P i
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