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

2.5.5.1.

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.