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2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.5. Propagation of error considerations

Formulas for functions of two variables

Case: Y=f(X,Z) Standard deviations of reported values that are functions of measurements on two variables are reproduced from a paper by H. Ku (Ku).

The reported value, Y is a function of averages of N measurements on two variables.

Function Y of Xbar, Zbar

Xbar andZbar are averages of N measurements

Standard deviation of Y

s(x) = standard dev of X;
s(z) = standard dev of Z;
s(xz)= covariance of X,Z

Note: Covariance term is to be included only if there is a reliable estimate

Y = A*Xbar + B*Zbar
Y = Xbar/Zbar
(1/SQRT(N))*(Xbar/Zbar)*SQRT{s(x)**2/(Xbar)**2 + s(z)**2/(Zbar)**2 - 
Xbar/(Xbar + Zbar)
(Xbar)**2*s(z)**2 + (Zbar)**2*s(x)**2 - 2*Xbar*Zbar*s(xz)**2
Y = Xbar*Zbar
s(x)**2/Xbar**2 + s(z)**2/Z**2 + 2*s(xz)**2/(Xbar*Zbar)}
Y = c*(Xbar)**a*(Zbar)**b
a**2*s(x)**2/X**2 + b**2*s(z)**2/Z**2 + 2*a*b*s(xz)**2/(Xbar*Zbar)}

Note: this is an approximation. The exact result could be obtained starting from the exact formula for the standard deviation of a product derived by Goodman (1960).

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