# Do you round uncertainties in intermediate calculations?

EDIT - As noted by Ben Crowell in the comments, my original answer here is not correct. Standard practice to reduce rounding error is to keep one (or two, if you want to be safe) extra digits in intermediate calculations. The intermediate uncertainty should certainly not have 10 significant figures, but 2 or 3 would be appropriate for safety.

Round it. The fact of the matter is that anything after the first (or arguably second) significant figure in an uncertainty estimate is meaningless. If you're uncertain of the value of the third digit after the floating point, then you obviously have no idea of the value of the eighth.

In Taylor's *Introduction to Error Analysis*, he writes

Several basic rules for stating uncertainties are worth emphasizing. First, because the quantity $\delta x$ is an estimate of an uncertainty, obviously it should not be stated with too much precision. If we measure the acceleration of gravity $g$, it would be absurd to state a result like $$\text{(measured g)} = 9.82 \pm 0.02385 \text{ m/s}^2$$The uncertainty in the measurement cannot conceivably be known to four significant figures. In high-precision work, uncertainties are sometimes stated with two significant figures, but for our purposes we can state that [experimental uncertainties should almost always be rounded to one significant figure].

[...] The rule has one significant exception. If the leading digit in the uncertainty $\delta x$ is a 1, then keeping two significant figures in $\delta x$ may be better. For example, suppose that some calculation gave the uncertainty $\delta x = 0.14$. Rounding this number to $\delta x = 0.1$ would be a substantial proportionate reduction, so we could argue that retaining two figures might be less misleading, and quote $\delta x = 0.14$. The same argument could perhaps be applied if the leading digit is a 2 but certainly not if it is any larger.

If you aren't familiar with the book, it's an outstanding resource for understanding how experimental errors propagate through calculations, and the meaning behind the statistical analysis that often gets left out of experimental science courses.