Question

How do you determine the percent error when adding, subtracting, multiplying, or dividing measurements, respectively?

Answer 1

Assuming uncorrelated variables, which is a good bet if you're in high school, the easiest thing to do would be to add fractional uncertainties. Example:
\[ A = 4 \pm 0.4\]
\[B = 10 \pm 2\]
The fractional uncertainty in A is
\[\frac{\sigma_A}{A} = \frac{0.4}{4} = 0.1\]
The fractional uncertainty in B is
\[\frac{\sigma_B}{B} = \frac{2}{10} = 0.2\]
Thus the fractional uncertainty in your answer will be their sum, i.e.
\[\frac{\sigma_A}{A} + \frac{\sigma_B}{B} = 0.1 + 0.2 = 0.3\]
This holds true for addition, subtraction, multiplication, or division. So,
\[A \cdot B = 40 \pm ?? \]
the fractional uncertainty is 0.3, and 0.3 * 40 is 12, so
\[A \cdot B = 40\pm 12\]
so on and so forth...

Answer 2

I use simple calculus... so if for example you're trying to find the error in the volumn of a sphere ... the formula for volumn is

\[V = (4/3)\pi*r ^{3}\]

and so if you're measuring r in cm to 1 mm than you have .1 possible error of measurement. You take the derivative of V

\[dV/dr = 4*\pi*r ^{2}\]

(dr) is your .1 and so dV is your error.. so

\[dV = 4*\pi*r^2*dr\]

Then later, if you want percentage of error you take dV and divide by V

Answer 3

^ as an FYI, the above method using differentials is exactly equivalent to adding fractional uncertainties, again assuming uncorrelated errors.

Answer 4

Thanks, Matt and Joe. Not only do I now know how to calculate error, I know the reasoning behind the calculations.