Some Notes on Elemen tary Statistical Consider-ations in MetrologyThe basic Measurement Model (display (2.1), page 19 of V&J) isy = x + where x is a "true value" of interest, is a measurement error, and y is whatis actually observed. We assume that is a random variable with mean β(the gauge "bias") and standard deviation σm easurement. (So under this model,repeat observation of the same x does not produce the same y.) Under thismodel, with x fixedEy = x + β and Vary = σ2m easurem entOn the other hand, with x random/varying and independent of xEy = µx+ β and Vary = σ2x+ σ2m easurem entFor a sample of m observations on the same unit with sample mean y and samplestandard deviation s,Ey = x + β and Es2= σ2m easurementand basic statistical methods can be applied to y and s to produce inferences ofmetrological interest. Consider first the usual confidence limits for a meany ± ts√m(t is based on m − 1 degrees of freedom.) These are limits for x + β.Ifthegauge is known to be well-calibrated (have 0 bias), they are limits for x,thesingle true value for the unit being measured. On the other hand, if x is knownbecause the unit being measured is a standard, it then follows that limits(y −x) ± ts√mcan serve as confidence limits for the gauge bias, β. Then consider the usualconfidence limits for a standard deviationÃss(m − 1)χ2m−1,up pe r,ss(m − 1)χ2m−1,lower!In the present context, these are limits for estimates σm easurement. Finally,mostly for purposes of comparison with other formulas, we might also note thata standard error for (an estimated standard deviation of) s isss12(m − 1)1For a sample of n observations, each on a different unitEy = µx+ β and Es2y= σ2x+ σ2m easurementApplying the usual confidence limits for a mean,y ± tsy√n(t is based on n − 1 degrees of freedom) are limits for µx+ β, the mean of thedistribution of true values for all units, plus bias. Note that the quantity sydoesn’t directly estimate anything of fundamental interest. But sinceσx=q(σ2x+ σ2m easurement) − σ2m easurementan estimate of unit-to-unit variation (free of measurement noise) based on asample of m observations on a single unit and a sample of n observations eachon different units is (see display (2.3), page 20 of V&J):bσx=qmax¡0,s2y− s2¢(*)The best currently available confidence limits on σxare complicated. But somereasonably elementary very approximate limits (based on what is known as "theSatterth waite approx imation") can be made. These areÃbσxsˆνχ2ˆν,up pe r, bσxsˆνχ2ˆν,lower!forˆν =bσ4xs4yn − 1+s4m − 1And it is possible to produce a “standard error” (an estimated standard devia-tion) for the estimate (*) as:bσxr12ˆν(we’re here ignoring the fact that these formulas can produce nonsense in thecase that bσx=0). These approximate confi dence limits and standard errorgive at least some feeling for how much one has really learned about σxbasedon the two
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