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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 ...
Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)
(Usually the expansion is done to first order; the second-order terms are needed to find the bias in the mean. Those second-order terms are usually dropped when finding the variance; see below). 5. With the expansion in hand, find the expected value. This will give an approximation for the mean of z, and will include terms that represent any bias.
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.
It is remarkable that the sum of squares of the residuals and the sample mean can be shown to be independent of each other, using, e.g. Basu's theorem.That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic:
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Standard addition involves adding known amounts of analyte to an unknown sample, a process known as spiking.By increasing the number of spikes, the analyst can extrapolate for the analyte concentration in the unknown that has not been spiked. [2]
The general form of the Eyring–Polanyi equation somewhat resembles the Arrhenius equation: = ‡ where is the rate constant, ‡ is the Gibbs energy of activation, is the transmission coefficient, is the Boltzmann constant, is the temperature, and is the Planck constant.