Standard deviation of a ratio and calculation of weight I have several "skilled" and "unskilled" wage observations for a number of countries, and would like to construct a single skilled-unskilled wage premium by country as the ratio of the simple average of skilled and unskilled wages.
Since individual wage reports are noisy, I would like to give more weight to wage premium estimates that have a reasonable number of observations in both categories in subsequent analyses. What would be a reasonable way of calculating the weight for each wage premium estimate? We can assume that the error terms of skilled and unskilled wage reports from a country are independently and normally distributed.
Some more considerations: If it was just one average wage, I think the central limit theorem suggests to weight each average by the square root of the number of observations that gave rise to it since Sd^(meanwage)=Sd^(individ. wage)/sqrt(N), and hence the standard deviation of the average decreases in the square root of N.
What is the analogous formula for Sd^(meanwage_skilled/meanwage_unskilled)? And given that it will likely be a little more complicated and involve the number of both skilled and unskilled wage observations, any suggestions on how to translate this into a weight for the wage premium?
Thanks!
 A: The easiest way to do this is to aggregate your data into an array. Have the first column represent the log of the weight and the second column represent a binary coding of 1: skilled vs. 0: unskilled labor status. Fit a regression model with log-weight as the outcome and labor status as the exposure. Take the coefficient and upper and lower 95% confidence limits and exponentiate all 3. These are interpreted as the ratio of weight comparing skilled to unskilled laborers. The p-value corresponds to the significance test that this ratio has a null value of 1 (no difference).
The central limit theorem doesn't apply to the ratio because you aren't sampling and thus averaging the ratios themselves. The "statistics" which are accruing are the skilled/unskilled laborer means. The $\delta$-method is another option for aggregating those data. Using the Jacobian matrix:
$$ \text{Var}\left( \bar{W}_{s}/\bar{W}_{u} \right) \approx 
\left[ \begin{array}{c} 1/\bar{W}_u \\ -(\bar{W}_s/\bar{W}_u)^2 \end{array} \right]^T 
\left[
\begin{array}{cc} \text{Var} \left( \bar{W}_{s} \right) & 0 \\
0 & \text{Var} \left( \bar{W}_{u} \right) \\
\end{array}
\right]
\left[ \begin{array}{c} 1/\bar{W}_u \\ -(\bar{W}_s/\bar{W}_u)^2 \end{array} \right]
$$ 
