Solved – How to derive the standard error of product of two variables with unequal sample sizes

Say $H$=height, $b$=angle, and $d$=distance; $H=text{tan}(b)*d$.

  • $b$ has a sample size of $5$ with mean $40.4166$ degrees and variance $3.75E-05$.

  • $d$ has a sample size of $3$ with mean $124.3$ and variance $0.01$.

Could you explain why $$text{tan}(b)^2*text{var}(H)+(H*sec(b)^2)^2*text{var}(b)=text{var}(text{tan}(b)*d)?$$ I understand using variance properties the first part but not the "$sec(b)^2$" part. Finally, how does one get from the $text{var}(text{tan}(b)*d)$ to the standard error of the estimation of $H$? I know standard error=stdev/sqrt(sample size) but I don't know what to do with multiple variables.

In general for a function of 2 variables f(x,y),

$$Var(f) approx (frac{partial f}{partial x})^2Var(x)+(frac{partial f}{partial y})^2Var(y)+frac{partial f}{partial x}frac{partial f}{partial y}Covar(x,y)$$.
Your formula is just an expression after evaluating the partial derivatives, but note that it has to be assumed that b and d are independent, since the 2 equations are equal only if the covariance is zero.

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