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

2.5.5.2.

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 \( \bar{X} \) , \( \bar{Z} \)

\( \bar{X} \) and \( \bar{Z} \) are averages of \(N\) measurements

Standard deviation of \(Y\)

\(s_x\) = standard deviation of \(X\)
\(s_z\) = standard deviation of \(Z\)
\(s_{xz}^2\) = covariance of \(X,Z\)

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

\( \large{ Y = A \bar{X} + B \bar{Z} } \)

\( \large{ \frac{1}{\sqrt{N}} \sqrt{A^2 s_x^2 + B^2 s_z^2 + 2AB s_{xz}^2} } \)

\( \large{ Y = \frac{\bar{X}}{\bar{Z}} } \)

\( \large{ \frac{1}{\sqrt{N}} \frac{\bar{X}}{\bar{Z}} \sqrt{\frac{s_x^2}{(\bar{X})^2} + \frac{s_z^2}{(\bar{Z})^2} - 2\frac{s_{xz}^2}{\bar{X} \bar{Z}}} } \)

\( \large{ Y = \frac{\bar{X}}{\bar{X} + \bar{Z}} } \)

\( \large{ \left( \frac{Y}{\bar{X}}\right)^2 \frac{1}{\sqrt{N}} \sqrt{(\bar{X})^2 s_z^2 + (\bar{Z})^2 s_x^2 - 2 \bar{X} \bar{Z} s_{xz}^2}} \)

\( \large{ Y = \bar{X} \bar{Z} } \)

\( \large{ \frac{\bar{X} \bar{Z}}{\sqrt{N}} \sqrt{\frac{s_x^2}{\bar{X}^2} + \frac{s_z^2}{\bar{Z}^2} + 2 \frac{s_{xz}^2}{\bar{X} \bar{Z}} } } \)

\( \large{ Y = c(\bar{X})^a (\bar{Z})^b } \)

\( \large{ \frac{Y}{\sqrt{N}} \sqrt{ a^2 \frac{s_x^2}{\bar{X}^2} + b^2 \frac{s_z^2}{\bar{Z}^2} + 2ab \frac{s_{xz}^2}{\bar{X} \bar{Z}} } } \)

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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