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

## Formulas for functions of one variable

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

The reported value, Y, is a function of the average of N measurements on a single variable.

Notes

Function $$Y$$ of $$\bar{X}$$

$$\bar{X}$$ is an average of $$N$$ measurements

Standard deviation of $$Y$$

$$s_x$$ = standard deviation of $$X$$

$$\Large{ Y = \bar{X} }$$

$$\Large{ \frac{1}{\sqrt{N}} s_x }$$

$$\Large{ Y = \frac{\bar{X}}{1+\bar{X}} }$$

$$\Large{\frac{s_x}{\sqrt{N} \left( 1 + \bar{X} \right)^2 } }$$

$$\Large{ Y = (\bar{X})^2 }$$

$$\Large{ \frac{2 \bar{X}}{\sqrt{N}} s_x }$$

$$\Large{ Y = \sqrt{\bar{X}} }$$

$$\Large{ \frac{s_x}{2\sqrt{N \bar{X}}} }$$

$$\Large{ Y = \mbox{ln} \bar{X} }$$

$$\Large{ \frac{s_x}{\bar{X} \sqrt{N}} }$$

Approximation could be seriously in error if N is small
$$\Large{ Y = e^{\bar{X}} }$$
$$\Large{ \frac{e^{\bar{X}}}{\sqrt{N}} s_x }$$
Not directly derived from the formulas

$$\Large{ Y = \frac{100}{\bar{X}} s_x }$$

$$\Large{ \frac{Y}{\sqrt{2(N-1)}} }$$

Note: we need to assume that the original data follow an approximately normal distribution.