Random and Systematic Errors

Two previous posts covered uncertainties in the context of thermal energy transfers but now we need to consider random and systematic errors as they apply more generally in the current AQA A-Level Physics course.

NOTE: The AQA A-Level Physics syllabus takes a simplified approach to uncertainties so if you are doing independent revision you may find online information that goes beyond the requirements of the current course. (This includes the earlier physbang posts, which are still available at; https://physbang.com/2023/11/12/experimental-errors/ and https://physbang.com/2023/11/26/experimental-errors-part-2/.)

A complete list of the exam board’s key terms can be downloaded from https://physbang.com/wp-content/uploads/2025/04/key-terms_aqa-7407-7408-phbk.pdf but the two important definitions for this post are discussed in detail below.

Random errors are unpredictable variations resulting in measurements that are equally likely to be distributed above and below the true value. Random errors cannot be eliminated but they can be reduced by taking repeated measurements and calculating the mean.

Random errors are particularly common when recording values of a quantity that is changing (such as the potential difference across a charging capacitor) or when the measuring point is subject to human judgement (such as when timing the swing of a pendulum). Parallax is also a common cause of random errors, as illustrated in the diagram below.

The uncertainty due to random errors is equal to +/- half the range of the results. For example, suppose the diameter of a wire is measured in different places along its length and the following values (in mm) are recorded;

The measurements go from 0.27 mm to 0.28 mm so the range is 0.01 mm and the uncertainty is therefore +/- 0.005 mm. The mean is 0.277 mm so the final value would probably be stated as 0.277 +/- 0.005 mm, which covers both of the measured values when rounded to three significant figures.

(There is a case for saying that the mean value should be rounded to the resolution of the original instrument, so the stated value ought to be 0.28 +/- 0.005 mm but the original data included no measurements above 0.28 mm so rounding the mean value wouldn’t truly reflect the original results – unless we regard the three 0.27 mm readings as anomalies, which is unlikely as they represent 30% of the total data.)

Systematic errors cause measurements to differ from the true value by a consistent amount every time. Compensation can be applied to eliminate systematic errors. Performing multiple measurements will not reduce systematic errors, which can be exposed by calibration or by using a different instrument or measurement technique.

The commonest systematic error is due to measurements not starting at zero. For example, a micrometer may not read zero when its jaws are fully closed; a newton meter may not read zero when no load is applied, a stopwatch needle may not return to zero when it is reset; a voltmeter may show a non-zero reading before it is connected across a component.

In all of these examples, the true value can be found by subtracting the zero error from the indicated measurement. So if a newton meter reads 0.01 N with no load applied and 5.32 N when measuring the weight of an object, the true weight must be 5.31 N. Similarly, if the zero reading had been -0.01 N then the true value would be 5.33 N (5.32 – -0.01).

The volume of liquid in a graduated (measuring) cylinder should always be determined by reading the position of the meniscus (the flat bottom area of the liquid surface) and if the upper reach of the liquid is used then a systematic error will arise. In both cases there may also be random errors due to human judgement when trying to locate the exact position against the scale on the side of the measuring cylinder.

There is a second type of systematic error, which causes the intervals to have the wrong value. The exam board seems less concerned with this type of error, which is easiest exposed by using a known reference. For example, it is common to state that a pan balance must be zeroed before any mass measurements are made in order to avoid a systematic error – and that is certainly important. But it is also necessary to use a reference mass to ensure that correct non-zero readings are obtained. For example, if the pan balance is zeroed but a 100 g reference mass produces a reading of 101.5 g then it is likely that all measurements will be too heavy by 1.5% and the appropriate correction must be applied to every reading.

Importantly, data readings should always be recorded in their raw form before being corrected: it is very bad practice to apply corrections as you are going along!

Ideally, there would be no systematic errors in the data as all instruments should be zeroed and adjusted to give correct calibration before use but the exam board expects you to state how you can eliminate systematic errors (by checking the zero reading, for example). There is no excuse for systematic errors that are due to poor technique, such as not measuring the meniscus position when using a graduated cylinder.

Random errors can never be eliminated completely as they have a minimum size that is determined by the resolution of the apparatus used. The exam board makes an important distinction depending on whether the instrument is digital or analogue; for digital read-outs, the minimum uncertainty is the +/- resolution of the apparatus whereas for analogue scales that are judged by eye, the minimum uncertainty is +/- half the resolution of the apparatus.

The resolution is the smallest division on the measurement scale so a digital voltmeter that reads to 0.01 V has a resolution of +/- 0.01 V whereas a rule that is marked with a millimetre scale has a limiting resolution of +/- 0.5 mm. It is worth noting that this is not a universally agreed standard; it is simply the exam board’s approach – and therefore the basis for correct answers in a current AQA A-Level Physics examination.

Finally, it is worth adding that, strictly speaking, uncertainty is a relative term and depends on the level of confidence. This takes us into the realms of statistical analysis, which is mostly beyond the A-Level Physics course. Nevertheless, you may see a value quoted as something like “6.32 kg +/-0.005 kg at a confidence level of 95%”. This means the uncertainty takes in 95% of the expected measurements, which corresponds to a statistical range of +/- two standard deviations.

For those who want more detail about uncertainties, the National Physical Laboratory has a collection of Measurement Good Practice Guides that can be downloaded from https://www.npl.co.uk/resources/gpgs. In particular, The Beginner’s Guide to
Uncertainty of Measurement (GPG11), by Stephanie Bell, is an easy read that covers all of the important facts: it can be downloaded from https://www.npl.co.uk/gpgs/beginners-guide-measurement-uncertainty-gpg11. There is also a very thorough summary by Ian Farrance and Robert Frenkel, entitled Uncertainty of Measurement: A Review of the Rules for Calculating Uncertainty Components through Functional Relationships, that can be read online or downloaded at https://pmc.ncbi.nlm.nih.gov/articles/PMC3387884/.