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Usual handling of measurement uncertainties

The present situation concerning the treatment of measurement uncertainties can be summarized as follows. The problem of interpretation will be treated in the next section. For the moment, let us see why the use of standard propagation of uncertainty, namely

$\displaystyle \sigma^2(Y) = \sum_i\left(\frac{\partial Y}{\partial X_i}\right)^2 \sigma^2(X_i) + {{\it correlation \, terms}}\,,$ (1.1)

is not justified (especially if contributions due to systematic effects are included). This formula is derived from the rules of probability distributions, making use of linearization (a usually reasonable approximation for routine applications). This leads to theoretical and practical problems. It is very interesting to go to your favourite textbook and see how `error propagation' is introduced. You will realize that some formulae are developed for random quantities, making use of linear approximations, and then suddenly they are used for physics quantities without any justification.1.6A typical example is measuring a velocity $ v\pm \sigma(v)$ from a distance $ s\pm \sigma(s)$ and a time interval $ t\pm \sigma(t)$. It is really a challenge to go from the uncertainty on $ s$ and $ t$ to that of $ v$ without considering $ s$, $ t$ and $ v$ as random variables, and to avoid thinking of the final result as a probabilistic statement on the velocity. Also in this case, an intuitive interpretation conflicts with standard probability theory.


next up previous contents
Next: Probability of observables versus Up: Uncertainty in physics and Previous: Sources of measurement uncertainty   Contents
Giulio D'Agostini 2003-05-15