AQA A Level Physics: Measurements and Their Errors — Revision Notes

Condensed for the final weeks. For the full explanation, use the Measurements and Their Errors study guide.

SI base units

Quantity	Unit
Mass	kilogram (kg)
Length	metre (m)
Time	second (s)
Current	ampere (A)
Temperature	kelvin (K)
Amount	mole (mol)

All other units are derived: N = kg m s⁻², J = kg m² s⁻², W = kg m² s⁻³, Pa = kg m⁻¹ s⁻².

Aside (not examined): dimensional analysis / homogeneity of units is not required by the AQA A-level specification and will not be examined, but as background: checking that the base units on each side of an equation match can catch some errors. Note a dimensionally consistent equation can still be wrong by a numerical factor.

Prefixes: T 10¹², G 10⁹, M 10⁶, k 10³, c 10⁻², m 10⁻³, μ 10⁻⁶, n 10⁻⁹, p 10⁻¹², f 10⁻¹⁵.

Accuracy, precision, resolution, repeatability, reproducibility

Term	Meaning
Accurate	Close to the true value
Precise	Repeat readings close to each other
Resolution	The smallest change in a quantity an instrument can detect
Repeatable	Same result on repeating with the same method and equipment
Reproducible	Same result with a different method or by a different person

Precise but not accurate is the key case. A miscalibrated instrument gives readings that agree closely with each other and are all consistently wrong. This distinction is examined nearly every series.

Errors

Random error — unpredictable scatter around the true value. Reduced by repeating and averaging, and by using instruments with finer resolution.

Systematic error — every reading is offset in the same direction by the same amount or proportion. Repeating does not help; the instrument must be recalibrated or zeroed. A zero error is the specific case where the instrument does not read zero when it should.

How to spot a systematic error on a graph: an additive systematic error (e.g. a zero error) gives the correct gradient but an unexpected intercept — a line that should pass through the origin but does not is the classic signature. A proportional (calibration) systematic error instead changes the gradient, while the intercept stays at the origin.

Uncertainty

uncertainty of a single reading (analogue scale)  =  half the smallest scale division
uncertainty of a single reading (digital display)  =  ± the resolution (1 unit of the last digit)
fractional uncertainty = uncertainty / value
percentage uncertainty = (uncertainty / value) x 100 = fractional uncertainty x 100

For a measurement requiring two readings — a length between two marks, a temperature change, a burette volume — the uncertainty doubles, because each reading carries its own.

Combining uncertainties:

Operation	Rule
Adding or subtracting	Add the absolute uncertainties
Multiplying or dividing	Add the percentage uncertainties
Raising to a power n	Multiply the percentage uncertainty by n

That last rule matters: if a radius is measured to 2% and you calculate a volume proportional to r³, the volume carries 6% uncertainty. Small measurement errors amplify quickly in cubed quantities.

To reduce percentage uncertainty, measure a larger quantity. The absolute uncertainty is fixed by the instrument, so it forms a smaller proportion of a bigger reading — which is why you time twenty oscillations rather than one, then divide.

Worked example. V = 4.85 ± 0.05 V and I = 0.42 ± 0.01 A. Find R with its uncertainty.

R = V/I = 4.85 / 0.42 = 11.55 Ohm
%unc(V) = (0.05/4.85) x 100 = 1.03%     %unc(I) = (0.01/0.42) x 100 = 2.38%
division -> percentages ADD: %unc(R) = 1.03 + 2.38 = 3.41%
absolute uncertainty = 0.0341 x 11.55 = +-0.39 Ohm
R = 11.6 +- 0.4 Ohm

The final answer is rounded to match the precision implied by the uncertainty — quoting more figures than that overstates how well the value is actually known.

Graphs

• Independent variable on x, dependent on y.
• Scales linear, using more than half the grid.
• Axes labelled with quantity and unit.
• Line of best fit, not dot-to-dot; identify and exclude anomalies.

Error bars show the uncertainty in each point. Draw the steepest and shallowest lines that pass through all the error bars:

uncertainty in gradient = (max gradient - min gradient) / 2
uncertainty in intercept = (max intercept - min intercept) / 2

Where error bars for two data sets overlap, the difference between them may not be significant.

Straight-line analysis: rearrange into y = mx + c form, then the gradient and intercept give the physical quantities you want. Identifying what to plot against what is usually the first mark.

Exam traps

• Swapping accuracy and precision.
• Saying repeating reduces systematic error.
• Forgetting to double the uncertainty for a two-reading measurement.
• Adding absolute uncertainties when multiplying.
• Forgetting to multiply the percentage uncertainty by the power.
• Quoting more significant figures than the data justifies.
• Drawing a line through the origin when the data does not support it.

Self-test

1. Distinguish accuracy from precision, and give an example of precise but inaccurate data.
2. Which type of error does averaging reduce, and which does it not?
3. How do you combine uncertainties when multiplying two quantities?
4. A radius is known to 3%. What is the uncertainty in a volume proportional to r³?
5. How is the uncertainty in a gradient and in an intercept found from a graph?

Answers: 1. Accuracy is closeness to the true value; precision is how closely repeated readings agree. A balance with a zero error gives tightly grouped readings that are all wrong by the same amount. 2. Averaging reduces random error; it does not reduce systematic error, which requires recalibration. 3. Add the percentage uncertainties. 4. 9% — the percentage uncertainty is multiplied by the power. 5. Draw the steepest and shallowest lines that still pass through all the error bars, then take half the difference between their gradients for the gradient uncertainty, and half the difference between their intercepts for the intercept uncertainty.