“Every measurement is subject to some uncertainty. A measurement result is only complete if it is accompanied by a statement of the uncertainty in the measurement. Measurement uncertainties can come from the measuring instrument, from the item being measured, from the environment, from the operator, and from other sources. Such uncertainties can be estimated using statistical analysis of a set of measurements, and using other kinds of information about the measurement process.”

— (Bell 1999, Abstract)

Bevington and Robinson (2002, 48) develop the propagation of uncertainties (errors) of multiple operations and functions. We provide the uncertainty propagation for a $\log_{10}$ function.

Starting from their general two variable formula $ y = f( u, v )$:

$$ \sigma_y^2 = \sigma_u^2 \left( \frac{\partial y}{\partial u} \right)^2 + \sigma_v^2 \left( \frac{\partial y}{\partial v} \right)^2 + 2\sigma_{uv}^2\left( \frac{\partial y}{\partial u} \right) \frac{\partial y}{\partial v} $$

Since $\log_{10}$ is a function of a single variable,

$$ \sigma_y^2 = \sigma_u^2 \left( \frac{d y}{du} \right)^2 $$

With $y = \log_{10}$ the derivitive of $y$ is

$$ \frac{d y}{du} = \frac{1}{u \ln(10)} $$

Yielding the propagation of the variance and standard deviation of the uncertainty of $y$ as:

$$ \sigma_y^2 = \sigma_u^2 \left( \frac{1}{u \ln(10)} \right)^2 $$

$$ \sigma_y = \sigma_u \left( \frac{1}{u \ln(10)} \right) $$

Let’s look at a counting experiment example.

$$ u = 736 \, \text{counts, } \sigma_u = \sqrt{u} = 27 $$

$$ y = \log_{10}( u ) = 2.87, \sigma_y = 0.02 $$

References

Bell, Stephanie. 1999

“Measurement Good Practice Guide No. 11 (Issue 2).” National Physical Laboratory. https://www.npl.co.uk/resources/gpgs/all-gpgs. https://www.npl.co.uk/special-pages/guides/gpg11_uncertainty.

Bevington, Philip R., and D. Keith Robinson. 2002

Data Reduction and Error Analysis for the Physical Sciences. 3rd ed. Science/Engineering/Math. Boston: McGraw-Hill. http://www.worldcat.org/oclc/865237466.

Kircher, Athanasius. 1679.

Coptic Letter Small Sigma. Image/png. File:Athanasius Kircher - Turris Babel - 1679 (page 232 crop - Coptic letter Sigma small).png. Wikimedia Commons, the Free Media Repository. https://upload.wikimedia.org/wikipedia/commons/2/2f/Athanasius_Kircher_-_Turris_Babel_-_1679_%28page_232_crop_-_Coptic_letter_Sigma_small%29.png.

Lenormand, Maxime, et al. 2012

“A Universal Model of Commuting Networks.” Edited by Renaud Lambiotte. PloS One 7 (October): e45985. https://doi.org/10.1371/journal.pone.0045985.