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 (96.245931)
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:
- 43.55 becomes 43.6
- 1.42500 becomes 1.42
- 1.42501 becomes 1.43 (because “501” is not exactly “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)