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While doing a physics lab, I noticed that the error analysis equation for multiplication $$R = \frac{X*Y}{Z}$$ $$ \delta R = |R|\sqrt{\left(\frac{\delta X}{X}\right)^2+\left(\frac{\delta Y}{Y}\right)^2+\left(\frac{\delta Z}{Z}\right)^2}$$ is not algebraically equivalent to $$R = X^n$$ $$\delta R = |n|*\frac{\delta X}{|X|}*|R|$$ For example, if we take $R = X*X$ then the first equation gives $\sqrt{2}*\left(\frac{\delta X}{|X|}\right)*X^2$. But if we take $R = X^2$ then the second equation gives $2*\left(\frac{\delta X}{|X|}\right)*X^2$.

Why is this the case? I have feeling that I should only use the first equation for differing variables $X$ and $Y$, but I don't understand why.

Robert Howard
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Ryan
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1 Answers1

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This comes down to correlation between $X$ and $Y$, and hence correlation in their errors.

These formulas for calculating $\delta X$ are in some sense talking about the standard deviation of the measurements of $X$.

If $X$ and $Y$ are independent measurements, then we don't expect the error in $X+Y$ to be just the sum of errors (it is too pessimistic) because there's some chance a positive error in $X$ will be cancelled out by a negative error in $Y$. Statistically you can then recover $\sqrt{\left(\delta X\right)^2+\left(\delta Y\right)^2}$ as the expected (average) error.

This reasoning completely fails when $X$ and $X$ are added. Whenever $X$ has a positive error, obviously $X$ (the second copy) also has positive error. There is no cancelling whatsoever, and the error of $2X$ is just twice the error of $X$.

obscurans
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