Random and systematic errors, precision and accuracy, repeatability and reproducibility, absolute, fractional and percentage uncertainty, and how uncertainties combine and are shown on graphs.

What this dot point is asking

AQA specification point 3.1.2 wants you to distinguish random from systematic errors, define precision and accuracy correctly, express uncertainty in absolute, fractional and percentage form, combine uncertainties through calculations, and represent uncertainty on graphs using error bars.

Random and systematic errors

Random errors come from unpredictable variations such as judging the exact moment to start a timer or reading a scale slightly differently each time. Systematic errors come from a fault in the method or apparatus, such as a ruler that has worn down at the zero end or a parallax error always made from the same viewing angle.

Precision and accuracy

These two words are not interchangeable in physics.

Repeatability is achieved when the same person, with the same method and equipment, gets consistent results. Reproducibility is achieved when a different person or a different method gives consistent results, which is a stronger test of reliability.

Expressing uncertainty

• Absolute uncertainty is the uncertainty in the same units as the measurement, for example $\pm 0.5 \text{ mm}$. For a single reading it is often taken as half the smallest scale division.
• Fractional uncertainty is $\dfrac{\text{absolute uncertainty}}{\text{measured value}}$.
• Percentage uncertainty is the fractional uncertainty multiplied by $100$.

To reduce percentage uncertainty, measure larger quantities (a longer time for many oscillations, a thicker stack of sheets) so the fixed absolute uncertainty is a smaller fraction of the value.

Combining uncertainties

Uncertainty on graphs

Plot points with error bars showing the absolute uncertainty in each value. The uncertainty in a gradient is found from the difference between the steepest and shallowest lines that still pass through all the error bars, often quoted as half that range. Error bars also reveal whether a point that lies off the line is genuinely anomalous or simply within its uncertainty.

Try this

Q1. A length is measured as $25.0 \pm 0.5 \text{ cm}$. State the percentage uncertainty. [1 mark]

• Cue. $\dfrac{0.5}{25.0} \times 100 = 2\%$.

Q2. Explain the difference between a random and a systematic error, and state how each is reduced. [3 marks]

• Cue. Random scatters readings and is reduced by repeating and averaging; systematic shifts all readings the same way and is reduced only by correcting the method or instrument.

Q3. State how the percentage uncertainty in a quantity raised to the power 3 relates to the percentage uncertainty in the quantity. [1 mark]

• Cue. It is three times as large.