1.2.1 Uncertainties and the ± Notation

You use a protractor to measure an angle. It looks closest to the 20° mark, but the true value may be anything between 19.5° and 20.5° (because of instrumental precision). You measure the thickness of a wire to be 2.0 mm, but it could be thicker or thinner at other points along the wire (because of manufacturing tolerance). You use a stopwatch to time the same event but each time you get a slightly different reading (because your reaction time is not constant). The reading on a very sensitive microbalance keeps fluctuating (because of tiny air currents).

For various practical reasons, we can never know the true exact value of what we’re trying to measure. This’s why measurements are often presented with their associated uncertainties using the ± notation.

For example,

$\displaystyle m=2.5\pm 0.1\text{ g}$

means that m is 2.5 g with an absolute uncertainty of 0.1 g.

Alternatively, we can present the fractional or percentage uncertainty

$\displaystyle \frac{{\Delta m}}{m}=\frac{{0.1}}{{2.5}}=0.04=4\%$

$\displaystyle m=2.5\text{ g }\pm 4\%$

There are pretty strict rules regarding the s.f. and d.p. when using the ± notation:

The absolute uncertainty must be rounded off to 1 significant figure only.
The measured value must be rounded off to the same decimal place (or place value) as the uncertainty.