I have an instrument of which I would like to understand the uncertainty on the measurements taken, so that every time that I perform a single measurement, I can apply the error obtained and therefore compare it to other measurements. In order to estimate the uncertainty, I took 30 measurements one after another.

The variable that I am interested in getting out of this instrument is actually the ratio of two variables, A and B, measured by the instrument at the same moment, so that for each round of measurement, I will have a pair (A$_i$, B$_i$), with i = 1, ... 30. My final variable therefore will be $y = \frac{A}{B}$.

What some people in my lab are doing is to take these 30 measurements, calculate the mean and the standard deviation of the ratio as if the ratio were a direct measurement. That is:

$\bar{y} = \frac{1}{N} \sum \frac{A_i}{B_i}$ and $\sigma_y = \frac{1}{N - 1} \sum (y_i - \bar{y})^2$

My first thought, though, would have been to treat A and B as two separate measurements (as they are), dependent to each other (since coming from the same instrument), and to calculate the standard deviation using the error propagation formula:

$\bar{y} = \frac{\bar{A}}{\bar{B}}$ and $\sigma_y = \sqrt((\frac{\partial y}{\partial A})^2 \sigma_A^2 + (\frac{\partial y}{\partial B})^2 \sigma_B^2 + 2 \frac{\partial y}{\partial A} \frac{\partial y}{\partial B} \sigma_{AB})$

Now my question: What is the theoretical difference between the two approaches? When should one use the first (taken that it is the correct approach), and when to use the second?


1 Answer 1


$\bar{y} = \frac{\bar{A}}{\bar{B}}$ isn't true. Small example with $n = 2, A = (1, 3), B= (1, 2)$ then $\bar y = 1.25$ but $\frac{\bar{A}}{\bar{B}} = 4/3 = 1.33...$. For the general point consider $C = \frac{1}{B}$ and compare the formula for the Covariance of $A$ and $C$.

If you had strong priors about $A,B$ then maybe it would make sense to model them, but if you just are trying to estimate mean and $\sigma$ of $y$ from just your sample, then you aren't gaining anything by not directly looking at $y$.

  • $\begingroup$ Could you please support this statement with some reference? Because if I look at any book for uncertainty calculation (for example, uscibooks.aip.org/books/…), then it is written that for any indirect measurement one should use the error propagation formula. That is where my confusion comes from. $\endgroup$
    – s.cerioli
    Commented Jan 25 at 8:59
  • $\begingroup$ Maybe I was a little fast on the draw after I saw the error in $\bar y$. I'm not quite an expert on error propagation. We yould test it with a simulation, but I'm not even sure waht you plug in for A,B when you only have a sample. $\endgroup$ Commented Jan 26 at 17:33

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