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In our lab notes, it states that for some relationship between variables of the form $Z=AB $, the errors are related by

$$(\Delta Z /Z)^2=(\Delta A/A)^2 + (\Delta B /B)^2$$

And that for the relation $Z=A^n$ the errors are

$$(\Delta Z /Z)=n(\Delta A/A)$$

However this appears inconsistent to me. If I take the first relation and replace B with A such that $Z=A^2$, I get an error

$$(\Delta Z /Z)^2=(\Delta A/A)^2 + (\Delta A /A)^2=2(\Delta A/A)^2$$

Yielding $(\Delta Z/Z)= \sqrt{2}(\Delta A/A)$

Which is not the same as $(\Delta Z/Z)=2(\Delta A/A)$ as predicted by the second relation.

Where am I going wrong?

dmckee --- ex-moderator kitten
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Meep
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1 Answers1

4

The normal rules of error analysis assume the errors in different quantities are uncorrelated. To use the error rule you cite for the product $AB$ one must assume that the errors for $A$ and $B$ are unrelated, which is not the case when $A=B$.

Update: added some text based on the comments.

There is some intuition here. The measured value for $A$ may be larger or smaller than the true value. When the errors are uncorrelated, then the same is true independently for $B$. Averaging over the different possibilities for both $A$ and $B$ separately leads to the $\sqrt{2}$. When the errors are correlated the averaging changes. If the measured value for $A$ is bigger than the true value, then both measured factors of $A$ will be larger by the same amount. Averaging over possibilities in this case leads to the factor of $2$. The fact that the errors in the uncorrelated case can be larger or smaller allows for cancellations when you average that do not occur in the correlated case, which is reflected in the fact that $\sqrt{2}<2$.

Andrew
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