Errors, uncertainties and error barsAQA A-Level Physics: Revision notes

A random error causes readings to scatter unpredictably above and below the true value, for example from reaction time or reading a scale by eye. It can be reduced by taking repeat readings and calculating a mean, or by using a larger sample or equipment with finer resolution.

A systematic error shifts all readings by the same amount or proportion in the same direction, for example a zero error, a wrongly calibrated instrument or always viewing a scale from the same wrong angle. Repeating does not remove it. Reduce it by checking the zero, calibrating against a known standard, correcting readings, or using a different method.

• Precision: how close repeated readings are to each other (small scatter).
• Repeatability: the same person, method and equipment give closely agreeing results.
• Reproducibility: different people or equipment, or a different method, give closely agreeing results.
• Resolution: the smallest change in the quantity that an instrument can detect (e.g. 0.01 mm on a micrometer).
• Accuracy: how close a reading is to the true value.

A set of readings can be precise but inaccurate if a systematic error shifts them all.

An uncertainty gives the range within which the true value is expected to lie.

• Absolute uncertainty: ΔX, with the same unit as X, e.g. 2.00 ± 0.02 cm.
• Fractional uncertainty: ΔX ÷ X.
• Percentage uncertainty: (ΔX ÷ X) × 100%.

From repeat readings, the uncertainty in the mean can be taken as half the range (largest minus smallest, divided by 2). For a single reading it is at least the instrument resolution.

The number of significant figures in a value should match its uncertainty: an uncertainty of ±0.02 cm goes with a value quoted to the nearest 0.01 cm, so write 2.00 ± 0.02 cm, not 2.0 ± 0.02 cm or 2.000 ± 0.02 cm. Give an uncertainty to 1 significant figure (2 if it starts with 1).

• Adding or subtracting quantities: add the absolute uncertainties. If a = 4.0 ± 0.2 cm and b = 6.0 ± 0.3 cm then a + b = 10.0 ± 0.5 cm, and b − a = 2.0 ± 0.5 cm.
• Multiplying or dividing: add the percentage uncertainties.
• Raising to a power: multiply the percentage uncertainty by the power. For a cube of side 2.00 ± 0.02 cm (1.0%), the volume has uncertainty 3 × 1.0% = 3.0%.

Worked example. Density = 85.0 g ÷ 8.00 cm³ with 0.59% in mass and 3.0% in volume. Total = 3.6%, so ρ = 10.6 g cm⁻³ ± 3.6% = 10.6 ± 0.4 g cm⁻³.

You do not combine uncertainties for trigonometric or logarithmic functions at this level.

An error bar on a data point shows the range of its uncertainty, vertically, horizontally or both. Individual points may or may not have error bars.

To find the uncertainty in a straight-line gradient:

• Draw a best-fit line through the points.
• Draw a worst acceptable line (steepest or shallowest) that still passes through all the error bars.
• Calculate both gradients using a large triangle.
• Uncertainty in gradient = difference between the best and worst gradients.

The intercept is treated the same way. If theory predicts that the line passes through the origin but the intercept range does not include zero, a systematic error is likely.