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Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a quantity measured on an interval or ratio scale.. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by a statement of the associated uncertainty, such as the standard deviation.
The probability that an uncertain number represented by a p-box D is less than zero is the interval Pr(D < 0) = [F(0), F̅(0)], where F̅(0) is the left bound of the probability box D and F(0) is its right bound, both evaluated at zero. Two uncertain numbers represented by probability boxes may then be compared for numerical magnitude with the ...
The above expression makes clear that the uncertainty coefficient is a normalised mutual information I(X;Y). In particular, the uncertainty coefficient ranges in [0, 1] as I(X;Y) < H(X) and both I(X,Y) and H(X) are positive or null. Note that the value of U (but not H!) is independent of the base of the log since all logarithms are proportional.
For example, an experimental uncertainty analysis of an undergraduate physics lab experiment in which a pendulum can estimate the value of the local gravitational acceleration constant g. The relevant equation [1] for an idealized simple pendulum is, approximately,
when the probability distribution of the value is known, it can be used to calculate an exact confidence interval; when the probability distribution is unknown, Chebyshev 's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
But if the accuracy is within two tenths, the uncertainty is ± one tenth, and it is required to be explicit: 10.5 ± 0.1 and 10.50 ± 0.01 or 10.5(1) and 10.50(1). The numbers in parentheses apply to the numeral left of themselves, and are not part of that number, but part of a notation of uncertainty.