measurement equals the same value of σ / μ · 100%. One step further is to say that the accuracy of the calibrated current meter is σ / μ · 100%. Usually this is only valid within a certain range of measurements as a different accuracy might apply to extreme low or high current. 2.3 E R RO R P RO PAG AT I O N F RO M M AT H E M AT I C A L R E L AT I O N S Often a required parameter is derived from other measurable parameters. For ex- ample, average flow in a river (under stationary and uniform conditions) can be derived from measuring stage, slope and roughness. Then the question arises how an error in the measured parameters translates into the error of the wanted parame- ter. This would not only determine the total error but also, which of the parameters is most critical to the total error and hence should be measured with more care. Relative simple equations can be derived at, giving the variance σ 2 of the wanted parameter expressed in terms of well known statistical terms (variance and co- variance) of the measured parameters. The co-variance determines whether or not measured parameters are statistical dependent. Suppose that, in order to find a value for the function q( x , y ), we measure the two quantities x and y . The standard error of σ 2 becomes: σ 2 q = ∂ q ∂ x 2 σ 2 x + ∂ q ∂ y 2 σ 2 y + 2 ∂ q ∂ x ∂ q ∂ y σ xy (2.1) This gives the standard deviation σ q , whether or not the measurement of x and y are independent, and whether or not they are normally distributed. If the measure- ments of x and y are independent the covariance σ xy will approach zero. With σ xy zero, the equation reduces to: σ 2 q = ∂ q ∂ x 2 σ 2 x + ∂ q ∂ y 2 σ 2 y (2.2) When the covariance σ xy is not zero we say that the errors in x and y are corre- lated. In this situation the uncertainty σ q in q( x , y ) is not the same as we would get from the formula for independent errors in x and y .
2.3 E R RO R P RO PAG AT I O N F RO M M AT H E M AT I C A L R E L AT I O N S 7 Equation 2.1 and 2.2 are the general rules for propagation of errors derived from mathematical relations. For a number of functions the relation for propagation of errors has been worked out below: 22.214.171.124 Example: Propagation errors 1 q ( x ) = ax + b (2.3) σ 2 q = a 2 · σ 2 x (2.4) 126.96.36.199 Example: Propagation errors 2 q ( x , y ) = a 1 x + a 2 y (2.5) For an independent relation between x and y : σ 2 q = a 2 1 · σ 2 x + a 2 2 · σ 2 y (2.6) For a dependent relation between x and y : σ 2 q = a 2 1 · σ 2 x + a 2 2 · σ 2 y + 2 a 1 a 2 σ xy (2.7) The maximum of the covariance σ xy is obtained with maximum correlation ρ = 1. In that case σ xy = σ x · σ y . This will be further explained in section 2.4 under regression correlation analysis. 188.8.131.52 Example: Propagation errors 3 q ( x , y ) = a · x b · y c (2.8) For an independent relation between x and y : σ 2 q = abx b - 1 y c 2 σ 2 x + acx b y c - 1 2 σ 2 y (2.9) Through defining the relative errors as follows, σ 2 q ¯ q 2 = r 2 q σ 2 x ¯ x 2 = r 2 x σ 2 y ¯ y 2 = r 2 y (2.10) this becomes: r 2 q = b 2 r 2 x + c 2 r 2 y (2.11)
8 T H E O RY O F E R RO R S 184.108.40.206 Example: Slope area method
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