Propagation of uncertainty

Question

I was taught that when adding/subtracting measurements, add the absolute uncertainties, and when multiplying/dividing measurements, add the relative uncertainties.

I have a problem in a lab where I need to find the difference between two measurements that are very close together, and are each very precise; however, when I propagate the absolute uncertainties, I get a huge relative uncertainty for the final result (upwards of 50%). Here it is:

(0.050 ± 5%) - (0.044 ± 3%) = ? = (0.050 - 0.044) ± ((0.050 * 5/100) + (0.044 * 3/100)) = 0.006 ± 0.00382

This ± 0.00382 corresponds to 63.7% relative uncertainty; am I doing something wrong? Both of the original measurements were pretty accurate (<12% RU). In this case can I just add their relative uncertainties instead of converting to absolute, which is very large in comparison with the difference between the measurements?

Answer 1

There is nothing wrong with the ±0.004 uncertainty you obtained. The relative uncertainty of the difference between close results is supposed to be far larger than that of the original measurements. One way of seeing why is stretching your scenario a bit: suppose you got 0.048 rather than 0.044 as the result of the second measurement. The difference, 0.002, would be small enough to be meaningless, as the second result would encroach into the uncertainty range of the first measurement. Your example is not so extreme; still, the large relative uncertainty is justified.

(Aside: an analogous effect can affect computer calculations done with limited precision. Suggestively enough, it is referred to as catastrophic cancellation).