Ratio Errors
Ratio: \[ r = \frac{N}{D} \] Error propogation:
\[ \sigma_r^2 = \frac{\partial r}{\partial N} \sigma_N\ \oplus \frac{\partial r}{\partial D} \sigma_D = \frac{1}{D} \sigma_N\ \oplus \frac{N}{D^2} \sigma_D \]
So \[ \sigma_r = \sqrt{\frac{\sigma_N^2}{D^2}\ + \frac{N^2}{D^4}\sigma_D^2 } \] Or \[ \frac{\sigma_r}{r} = \sqrt{\frac{D^2}{N^2}\frac{\sigma_N^2}{D^2}\ + \frac{D^2}{N^2}\frac{N^2}{D^4}\sigma_D^2 } \]
Thus… \[ \left(\frac{\sigma_r}{r}\right)^2 = \left(\frac{\sigma_N}{N}\right)^2 + \left(\frac{\sigma_D}{D}\right)^2 \]
Product of an arbitrary number of factors
If \[ f = C_1^{e_1} C_2^{e_2}... C_N^{e_1N} \]
Then
\[ \frac{\partial f}{\partial C_i} = e_i \frac{f}{C_i} \]
And
\[ \frac{\sigma_f}{f} = \frac{1}{f}\sqrt{ \sum_i \left(e_i \frac{f}{C_i}\right)^2 \sigma_{C_i}^2\ } = \sqrt{ \sum_i \left( \frac{e_i}{C_i}\right)^2 \sigma_{C_i}^2\ } \]
Thus… \[ \left(\frac{\sigma_f}{f}\right)^2 = \sum_\oplus\frac{e_i^2 \sigma_{C_i}}{C_i} \]
In the simple ration above \(e_i = \pm 1\) and \(e_i^2 = 1\).
Follow-ups
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