The uncertainty associated with the weighted mean of measurement data

Abstract

Weighted mean has been used to combine means from several sets of measurements, which are either from different laboratories or based on different measurement methods. It is also often used in key comparisons and other inter-laboratory comparisons. However, the traditional estimator of the variance of the weighted mean underestimates the variance. Two new variance estimators are proposed with smaller biases, smaller mean square errors and much better coverage probabilities for the correspondingly formed intervals. In metrology, the uncertainties based on Type B evaluation which are not based on statistical analysis of data cannot be ignored. In the later part of this paper, the results are extended to the general case with both Type A and Type B uncertainty components being presented.

Incorporating ecological drivers and uncertainty into a demographic population viability analysis for the island fox

“…Thus, to estimate process variance, the directly estimated temporal variance should be discounted for the effects of sampling variance (mean rates also require adjustment; see Kendall 1998, White 2000, Morris and Doak 2002 for further discussion). We used random effects models in Program MARK to estimate the age-(i ) and grid-( j ) specific process variance, G ij using the global model (White 2000, White et al 2001, Burnham and White 2002; we summarized the overall global process variance for each age class, G i , as the mean of these separate grid-specific estimates weighted by the inverse of the sampling variance of the estimate for each grid (Zhang 2006). To ensure that the population models based on ecological drivers accounted for the full global process variance, we estimated the unassigned age-specific temporal process variance, U i , as the weighted average difference (Zhang 2006) between the process variance estimates of the global model, G ij , and those of the covariate models, C ij .…”

“…For the purposes of the PVA model, we expressed each unassigned process variance, U ij , as a proportion of the maximum possible variance, which for a survival rate is set by the mean,S (i.e.,S[1 ÀS ]) (Morris and Doak 2004; see Simulation methods: Adding unassigned process variation). Our overall estimate of V u (unassigned process variance as a proportion of the maximum possible variance) for each age class was the mean of the grid-specific V u values, weighted by the inverse of the sampling variance of the estimate for each grid (Zhang 2006). Process covariance between agespecific survival rates was imposed by covariance in the fitted coefficients of the logistic functions that predict survival rates.…”

“…Component variances were estimated using standard theory (i.e., var(S ) [White 2000]; var(G), var(C) [Doak et al 2005]). Variance of the overall V u estimate for each age class was estimated as the inverse of the sum of the inverses of these grid-specific variances (Zhang 2006).…”

Application of quantile functions for the analysis and comparison of gas pressure balance uncertainties

“…These limitations of Graybill-Deal estimators in the case of a single measurand are only applicable in the context of the simplest pressure balance model to the extent that the perfectly straight piston/cylinder geometry assumption is valid, however in a practical context it is well known that there are generally fluctuations in the piston/cylinder radii profiles along the pressure balance's engagement length and as a result the assumption of single effective values of ⟨r⟩ and ⟨R⟩ is only an approximation. Whilst more refined estimators have been developed the validity of the use of these alternative estimators assumes a large number of independent laboratory measurements for the underlying data, and as a result the use of Graybill-Deal estimators may not necessarily be considered a reasonable practical first approximation for ⟨r⟩ and ⟨R⟩, and we comment that when access to independent repeat dimensional measurements is available that alternative higher accuracy formulations may be used as discussed by Zhang [2] and other researchers.…”

“…Noting that h = Q(r) and r = G(h) we can simply use the previous cumulative distribution functions to determine the values u p i ¼ Qðp i Þ for i ∈ [1,2,3,4]. When this is implemented we then have three sets of non-linear equations specified as ½p…”