2.
Measurement Process Characterization
2.5. Uncertainty analysis 2.5.5. Propagation of error considerations
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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 |
\( s_x \) = standard deviation of \(X\) |
\( \Large{ Y = \bar{X} } \)
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\( \Large{ \frac{1}{\sqrt{N}} s_x } \)
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\( \Large{ Y = \frac{\bar{X}}{1+\bar{X}} } \)
| \( \Large{\frac{s_x}{\sqrt{N} \left( 1 + \bar{X} \right)^2 } } \)
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\( \Large{ Y = (\bar{X})^2 } \)
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\( \Large{ \frac{2 \bar{X}}{\sqrt{N}} s_x } \)
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\( \Large{ Y = \sqrt{\bar{X}} } \)
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\( \Large{ \frac{s_x}{2\sqrt{N \bar{X}}} } \)
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\( \Large{ Y = \mbox{ln} \bar{X} } \)
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\( \Large{ \frac{s_x}{\bar{X} \sqrt{N}} } \)
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Approximation could be seriously in error if N is small | ||
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. |