8.3, 3.1.2Required Practicals and Data AnalysisCore

Paper 3: Data Analysis and Uncertainty

A strong practical answer links a specific limitation to its effect on the result and to a workable improvement. Use gradients, logarithms and uncertainty intervals to make that argument quantitative.

40 min 8.2–8.3 Required Practicals and Data Analysis

What you'll be able to do

Distinguish accuracy, precision and systematic error
Propagate simple measurement uncertainties
Extract constants and test models with graphs

Choose the uncertainty model

A random effect varies unpredictably between readings and contributes scatter. Repeating independent readings and averaging can reduce its effect on a mean. A systematic effect, such as a zero error, biases results in a consistent way and is not removed simply by repeating.

Absolute uncertainty has the same unit as the reading; fractional uncertainty is their ratio; percentage uncertainty multiplies that ratio by 100. A digital resolution is not automatically the full uncertainty: response time, alignment, calibration and the physical method can dominate.

Propagate and interpret

For the usual worst-case A-level estimate, add absolute uncertainties for sums or differences. For multiplication or division add fractional uncertainties. A power multiplies a fractional uncertainty by the magnitude of the exponent. State the approximation: a rigorous independent-random-uncertainty calculation would use a different combination rule.

Quote uncertainty with sensible significant figures and align the value’s decimal place to it. Two results may be consistent if their plausible intervals overlap; a percentage difference alone does not establish disagreement. A much more precise-looking calculator display does not improve the underlying measurement.

y=\frac{ab^2}{c}\quad\Rightarrow\quad \frac{\Delta y}{|y|}\approx\frac{\Delta a}{|a|}+2\frac{\Delta b}{|b|}+\frac{\Delta c}{|c|}

This is the common worst-case small-uncertainty approximation, not a universal statistical identity.

Make a graph answer the physics question

Choose axes that linearise the model: y=kxⁿ gives ln(y/yref)=n ln(x/xref)+constant; an exponential decay gives a straight line when log response is plotted against time. Label quantities and units, use a large gradient triangle on the fitted line and avoid calculating a slope from two neighbouring raw points.

For an uncertainty in gradient, compare the best fit with plausible steepest and shallowest lines consistent with the error bars. Interpret the intercept physically before forcing a line through zero. An improvement should name the cause and explain how it reduces the relevant uncertainty, for example a longer timing interval to reduce fractional reaction-time error.

Work through the reasoning

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.

Example 1A measured radius is

(10.0\pm0.2)

mm. Estimate percentage uncertainty in area πr².

1Percentage radius uncertainty

=0.2/10.0\times100=2\%

2Area depends on r², so multiply by two.

AnswerApproximately 4%.

Example 2A decay graph of ln(V/1 V) against t has best slope −0.20 s⁻¹ and plausible slope range −0.22 to −0.18 s⁻¹. Estimate τ and a range.

\tau=-1/m=5.0

2The limiting values are

1/0.22=4.55

s and

1/0.18=5.56

3The reciprocal creates a slightly asymmetric interval.

Answerτ≈5.0 s with a plausible range about 4.5–5.6 s.

Equation recap

y=\frac{ab^2}{c}\quad\Rightarrow\quad \frac{\Delta y}{|y|}\approx\frac{\Delta a}{|a|}+2\frac{\Delta b}{|b|}+\frac{\Delta c}{|c|}

This is the common worst-case small-uncertainty approximation, not a universal statistical identity.

Common mistakes to avoid

Calling precise repeated measurements necessarily accurate.

They may share a systematic bias.

Giving an improvement without explaining its effect.

Connect it to the dominant uncertainty or model limitation.

Key takeaways

Match uncertainty statements to the measurement method.
Use the correct propagation approximation.
Graphs should extract a constant or test a stated physical model.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

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