This is a topic which seems to confuse a lot of students but it needn’t be so. The word error suggests a mistake in a result but this is not the case: the word uncertainty is clearer: any measurement taken in a practical or experiment has a margin of uncertainty associated with it.

For example, if you use a balance to find the mass of a stone to be 154g, that is a measurement taken to the nearest gram. Assuming there is no actual mistake in your measurement, the true mass of the stone could be anything from 153.5g to 154.5g. There are other sources of uncertainty which we will discuss in due course.

Any calculation of uncertainty in a measurement is really an estimate. The calculations involved in A Level physics are relatively simple. If you go on to study physics (or engineering or maths) at university, you will learn more involved but more precise estimating techniques.

Some of these methods are pretty much universal, some can vary slightly by awarding body (“exam board.”) It is always worth checking the specific methods used by your exam board.

First, a little house-keeping: uncertainty does not mean a mistake in a reading, anomalous readings are to be discarded and where possible repeated.

For a single reading:

Wherever possible, it is better to take repeat readings and use a mean value in your calculations. However, this is not always possible, for example when testing a wire to destruction in order to find its breaking stress.

The margin of uncertainty in a single reading is taken to be half the smallest increment on the measuring scale. This is because any reading should be rounded to the increments given on the scale. For example, if you measure the length of a pencil as 15cm to the nearest cm, the true length could be anywhere between 14.5 and 15.5cm.

NB: For small scales such as millimetres, some exam syllabuses used to take the uncertainty to be the whole of the increment on the measuring scale. This is because with a small scale like mm, it may be possible to inadvertently round the wrong way as the scale is difficult to read by eye. At the time of writing, all major exam syllabuses used a half increment as the uncertainty in a single reading. You are advised to check with your teachers or directly with your exam board which method they are currently using.

Another point to note: if you are measuring a length with a ruler, you should double the uncertainty because it’s possible to incur an uncertainty at either end of the ruler. (Unless you have some way of clamping the object you are measuring against the ruler on a flat surface.)

Absolute uncertainty

[Reading] ± [½ increment]

Percentage uncertainty

([½ increment] ÷ [reading]) x 100%

For Repeat Readings.

Wherever possible, you should take repeat readings and use the average in your calculations (having discarded and ideally repeated any anomalous readings.) The uncertainty in this average is then calculated by:

Absolute uncertainty

[Average] ± [ ½ range]

Where Range = Largest – Smallest value

Percentage uncertainty

([ ½ range] ÷ [Average]) x 100%