# Calculate mean and standard deviation of the ratio of two dependent variables

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?

$$\bar{y} = \frac{\bar{A}}{\bar{B}}$$ 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 $$y$$.
• 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. Commented Jan 26 at 17:33