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 $$\bar{X}$$ , $$\bar{Z}$$ $$\bar{X}$$ 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).