By Barendregt H., Barendsen E.
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28), putting however x ¯i − ud¯i ≤ β¯ ≤ x ¯i + ud¯i ; i = 1, . . , m . ¯ The lower diagrams confront the x ¯i ± ud¯i ; i = 1, . . , m with β. Though the grand mean promises more stability than the individual means, the uncertainties ud¯i prove intricate and somewhat sensitive constructions as they are interdependent. For an illustration, let us consider two further examples, Figs. 5. Even if the results x ¯i ±ux¯i ; i = 1, . . , m mutually overlap, it may happen that one mean or even more means abstain from localizing the true value x0 .
2 Uncertainty and True Value Let there be n repeated measurements of some measurand. The simplest least squares estimator of the true value x0 is known to be the arithmetic mean x ¯= 1 n n xl . 1) l=1 The presence of systematic errors causes the mean to be biased. To that eﬀect, the uncertainty ux¯ should be apt to localize the true value x0 of the measurand, x ¯ − ux¯ ≤ x0 ≤ x ¯ + ux¯ . In general, arithmetic means aiming at one and the same physical quantity and coming from diﬀerent laboratories will not coincide.
50 6 Means and Means of Means Fig. 3. Check of consistency of results x ¯i ± ux¯i ; i = 1, . . 4 Individual Mean Versus Grand Mean 51 Fig. 4. Though x ¯i ± ux¯i ; i = 2, 3 fail to localize x0 , none of the uncertainties ud¯i ; i = 1, . . , 5 beckons inconsistency 52 6 Means and Means of Means Fig. 5. Under additional modiﬁcations, the same means x ¯i ± ux¯i ; i = 2, 3 continue to fail to localize x0 . Now, however, the uncertainties ud¯1 and ud¯5 beckon inconsistency 7 Functions of Erroneous Variables To assess the inﬂuence of measurement errors propagated via functions, we conﬁne ourselves to linearized series expansions.