OxfordAQA 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 units and homogeneity

Base units: kg, m, s, A, K, mol.

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 is not required by this specification. For background only: checking homogeneity means confirming both sides of an equation have the same base units. Even a homogeneous equation may still be wrong by a dimensionless constant, so homogeneity shows an equation could be right, not that it definitely is.

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

Term	Meaning
Accurate	Close to the true value
Precise	Repeat readings agree closely
Resolution	Smallest change the instrument can detect
Repeatable	Same result, same method and operator
Reproducible	Same result, different method or operator

Precise but not accurate is the case that carries marks: a miscalibrated instrument produces tightly clustered readings that are all wrong by the same amount.

Higher resolution does not guarantee accuracy — a micrometer with a zero error reads to 0.01 mm and is still wrong every time, consistently offset from the true value.

Errors

Random — scatter about the true value; reduced by repeating and averaging and by higher-resolution instruments.

Systematic — a consistent offset in the same direction every time; not reduced by repeating. Requires recalibration, zeroing, or a technique change.

On a graph: an additive systematic error (e.g. a zero error) gives the correct gradient with an unexpected intercept, while a proportional (calibration) systematic error instead changes the gradient and leaves the intercept at the origin. Random error shows as scatter of points about the line, in both directions.

Uncertainty

reading uncertainty  =  half the smallest scale division
two-reading measurement  =  double it
percentage uncertainty = (uncertainty / value) x 100

Operation	Combine
Add / subtract	Absolute uncertainties add
Multiply / divide	Percentage uncertainties add
Power n	Percentage × n

The power rule is the one most often forgotten, and it matters most: a length measured to 2% gives a volume uncertain to 6%.

Reduce percentage uncertainty by measuring more — 20 oscillations rather than 1, a stack of 50 sheets rather than 1. The absolute uncertainty is set by the instrument, so it becomes a smaller proportion of a larger measured quantity.

Worked example

A wire has length 0.850 m ± 0.001 m and diameter 0.36 mm ± 0.01 mm. Find the percentage uncertainty in the cross-sectional area.

Area A = pi d^2 / 4, so A depends on d squared.

% uncertainty in d = (0.01 / 0.36) x 100 = 2.8%
% uncertainty in A = 2 x 2.8% = 5.6%

The diameter dominates the uncertainty: the length is known to about 0.1%, so improving the ruler would be pointless while the micrometer reading is this uncertain. Identifying the dominant uncertainty is usually the mark-earning comment in an evaluation question.

Graphical analysis

Rearrange to y = mx + c so gradient and intercept yield the physical quantities. Deciding what to plot is usually the first mark.

• Linear scales using more than half the grid.
• Axes labelled with quantity and unit.
• Best-fit line, anomalies identified and excluded.

Error bars: draw the steepest and shallowest lines consistent with all the bars.

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

Overlapping error bars between two data sets mean the difference between them may not be statistically significant, and this comparison is a common evaluation question.

Estimation

Order-of-magnitude estimates are examinable: state assumptions, use round numbers, and give the answer to one significant figure. The marks are for a reasoned method, not precision — so show the assumption explicitly, since a plausible but unstated assumption earns nothing.

Exam traps

• Swapping accuracy and precision.
• Assuming high resolution implies accuracy.
• Saying repeating reduces systematic error.
• Forgetting to double uncertainty for two-reading measurements.
• Forgetting the power rule.
• Quoting more significant figures than the raw data supports.

Self-test

1. Distinguish resolution from accuracy.
2. Which error type does averaging reduce?
3. A length is known to 2%. What is the uncertainty in a volume proportional to L³?
4. How do you find the uncertainty in a gradient?
5. A wire has length 0.850 m ± 0.001 m and diameter 0.36 mm ± 0.01 mm. Find the percentage uncertainty in its cross-sectional area, and identify the dominant source.

Answers: 1. Resolution is the smallest change an instrument can detect; accuracy is closeness to the true value — a high-resolution instrument with a zero error is precise and high-resolution but inaccurate. 2. Random error only. 3. 6%. 4. Draw the steepest and shallowest lines that pass through all the error bars and take half the difference between their gradients. 5. % uncertainty in d = (0.01 ÷ 0.36) × 100 = 2.8%; since A ∝ d², % uncertainty in A = 2 × 2.8% = 5.6%. The diameter dominates, since the length is known to only about 0.1%.