I have a physical system where a real and positive quantity A can be measured by two ratios between 4 measures, $\frac{B}{C}$ and $\frac{D}{E}$ (all of them are positive real numbers).
From the expressions I got from error propagation I'm having the sensation that it doesn't make a difference to either multiply these ratios and get the square of the product or just average both.
Making $X=A²=\frac{BD}{CE}$ first, then $$dX/X = \big[(dB/B)²+(dC/C)²+(dD/D)²+(dE/E)²\big]^{(1/2)}$$
So $$dX=\frac{BD}{CE}*\big[(dB/B)²+(dC/C)²+(dD/D)²+(dE/E)²\big]^{(1/2)}$$
Now, in the square root:
$$A=sqrt(X) \rightarrow \frac{dA}{A} =\frac{1}{2}\frac{dX}{X} = \frac{1}{2}\bigg(\frac{BD}{CE}\bigg)\frac{\bigg[(dB/B)²+(dC/C)²+(dD/D)²+(dE/E)²\bigg]^{(1/2)}}{\bigg(\frac{BD}{CE}\bigg)}$$
$$dA = \frac{1}{2}A[(dB/B)²+(dC/C)²+(dD/D)²+(dE/E)²]^{(1/2)}$$
Now, if instead I took the average of their ratios, first I need to find the errors of these ratios:
$$X=\frac{B}{C} \rightarrow dX = \bigg(\frac{B}{C}\bigg)*[(dB/B)²+(dC/C)²]^{(1/2)}$$ $$Y=\frac{D}{E} \rightarrow dY = \bigg(\frac{D}{E}\bigg)*[(dD/D)²+(dE/E)²]^{(1/2)}$$
$$A=(X+Y)/2 \rightarrow dA = \frac{1}{2}*\bigg(\bigg[\big(\frac{B}{C}\big)*[(dB/B)²+(dC/C)²]^{(1/2)}\bigg]^2+\bigg[\big(\frac{D}{E}\big)*[(dD/D)²+(dE/E)²]^{(1/2)}\bigg]^2\bigg)^{(1/2)}$$
As $\frac{B}{C}$ and $\frac{D}{E}$ are also equal to A, then this reduces to the same expression as before
$$dA = \frac{1}{2}A[(dB/B)²+(dC/C)²+(dD/D)²+(dE/E)²]^{(1/2)}$$
Is this correct or am I missing something?
