An important principle of uncertainty is that *results calculated from a measurement are at least as uncertain as the measurement itself*. In other words, you cannot gain precision in a calculation – you can only become less precise as more measurements are combined. Take the uncertainty in measurements into account to avoid misrepresenting the uncertainty in calculated results.

Raw data and the *final *reported value should be written with only **one uncertain or estimated digit** – do not include any values past this final significant figure. *Intermediate *calculations can – and should – contain additional uncertain digits (at least two) in order to reduce errors caused by rounding during the calculations. You should indicate these ‘extra’ digits in a way that lets you track the correct number of significant figures during your calculations.

You can:

- Underline the last significant digit in the value (e.g. 96.245931 for a number that should have five significant figures)
- Use subscript for the ‘extra’ digits (96.245
_{931})

## Rounding Your Final Answer

When you report your final calculated answer, you should write only the significant digits in the value – here is where you finally round off your value. For most numbers, the rules will be familiar:

- If the first digit to be dropped (the one immediately to the right of the last significant digit) is
**between 0 and 4**(inclusive),*leave the last significant digit unchanged*. (i.e. 83._{4}becomes 83) - If the first digit to be dropped is
**between 6 and 9**(inclusive),*increase the last significant digit by 1*. (i.e. 83._{6}becomes 84)

**If the first digit to be dropped is a 5**, then you may want to follow an additional rule in order to make sure no bias (accuracy error) is introduced in your value:

- If any nonzero digits follow the 5, then round up (i.e. 2.
_{52}rounds up to 3) - If the 5 is the last digit available or is followed by only zeros, round either up or down to produce an
*even*number in the last remaining digit:- 43.5
_{5}becomes 43.6 - 1.42
_{500}becomes 1.42 - 1.42
_{501}becomes 1.43 (because “501” is not exactly “5”!)

- 43.5

This rule may seem *odd *– but it helps avoid the situation where all numbers existing exactly in the middle of our two choices get rounded up. While this would be a small error overall, over several calculations, the error builds and your final answer could be higher than it should be just because of rounding bias.

**Remember – **most rounding error can be eliminated simply by carrying extra digits through your calculations, and *only *rounding off when presenting your final answer! (*this doesn’t actually eliminate the rounding error in each step, but pushes it into the digits that will be dropped at the end anyway*)