# Error propagation for ratio data

I'm working with some published ratio data for isotopes of uranium (238U) and lead (208Pb and 206Pb). The data are published in ratios of 238U/208Pb and 206Pb/208Pb, but I need the data to be in the form 238U/206Pb, which is simple enough to do. 238U = a, 208Pb = b and 206Pb = c:

(a/b) / (c/b) = a/c

Therefore:

(238U/208Pb) / (206Pb/208Pb) = 238U/206Pb

Each of the original isotope ratios has an associated error value, which is given as an absolute value at the 1 sigma level. Using the available data e.g. ratio and error values, would it be possible to calculate the associated error for the 238U/206Pb ratio, and if so, how would one go about doing this?

Here's an example of some data, I've also included a ρ (correlation coefficient for the original ratio data in case it's of any use.

206Pb/208Pb 1σ abs 238U/208Pb 1σ abs ρ 1.235 0.054 1652 119 0.9912

Many thanks, Chris

## 1 Answer

Consider your expression as $f(u, v)= u/v$, expand by Taylor's theorem: $$f(u_0 + a, v_0 + b) = f( u_0, v_0) + f_u( u_0, v_0) a+ f_v( u_0, v_0) b + \ldots$$ Here $f_u( u_0, v_0)$ is the partial derivative with respect to $u$. Under the assumption that the above linear relation is correct, and that the errors are independent, it will give you the standard deviation of the error from the standard deviations of the errors in $u$ and $v$. You can continue the expansion to include cross terms to account for correlations, but that gets messy real fast.