Significant Figures: Addition is too precise
Saturday December 11, 2021
The usual rules for adding and subtracting numbers with significant digits often propagate uncertainly approximately correctly, but always give results more precise than they actually are because they don't (and can't, generally) follow the Variance Sum Law.
Say a number in Significant Figures with rightmost significant digit \( D \times 10^N \) has uncertainty with standard deviation \( \sigma \times 10^N \), and assume errors are always uncorrelated.
So the number 12.3, with three significant figures, has uncertainty \( \sigma \times 0.1 \), and 2.48 has \( \sigma \times 0.01 \). Adding them gives 14.8, which has the same uncertainty as 12.3. By the Variance Sum Law, the true uncertainty is \( \sigma \times 0.1005 \), but that's pretty close to \( \sigma \times 0.1 \). In this way, the usual rule for adding with significant figures is often reasonable-seeming.
With many numbers of the same precision, however, the usual rules are more problematic. If you add 1.2 + 3.4 + 5.6 + 7.8, the result 18.0 implies \( \sigma \times 0.1 \), but in fact uncertainty has doubled to \( \sigma \times 0.2 \). Significant Figures has no way to convey this, because it only communicates in powers of ten.
Adding and subtracting 100 numbers with the same precision, then, should give a result with exactly one fewer significant figures. With 25 numbers the standard deviation could “round up” to the next power of ten, arguably. It may not be common to add so many numbers with significant figures, but even with just a few, Sig Figs is a course approximation of correct propagation of uncertainty.