What are errors and uncertainties in the context of Cambridge International A Levels Physics?

In Cambridge International A Levels Physics, errors and uncertainties refer to the potential deviations in measurements from the true value. Errors can be systematic, caused by flaws in measurement instruments or methods, or random, arising from unpredictable variations. Uncertainties quantify the doubt in measurements, providing a range within which the true value is expected to lie.

How do systematic errors differ from random errors in physics experiments?

Systematic errors are consistent and repeatable inaccuracies that occur due to faulty equipment, calibration errors, or flawed experimental design, leading to a bias in measurements. Random errors, on the other hand, are unpredictable and vary in magnitude and direction, often caused by environmental factors or human limitations, and can be reduced by taking multiple measurements and averaging the results.

Why is it important to consider uncertainties in physics measurements?

Considering uncertainties in physics measurements is crucial because it provides a realistic range of values within which the true measurement is likely to fall. This helps in assessing the reliability and precision of experimental results, comparing them with theoretical predictions, and ensuring that conclusions drawn from the data are valid and scientifically sound.

How can one calculate the uncertainty in a measurement?

To calculate the uncertainty in a measurement, one can use methods such as the absolute uncertainty, which is the margin of error in the same units as the measurement, or the percentage uncertainty, which expresses the uncertainty as a percentage of the measured value. For multiple measurements, the standard deviation can be used to estimate the uncertainty, providing a statistical measure of the spread of values.

What role do significant figures play in expressing uncertainties in physics?

Significant figures play a crucial role in expressing uncertainties as they indicate the precision of a measurement. The number of significant figures reflects the certainty of the measurement, with more significant figures suggesting higher precision. When reporting uncertainties, it is important to match the number of significant figures in the uncertainty to the least precise measurement to maintain consistency and accuracy.

Why is "Adding absolute uncertainties for products" a common mistake in Errors and uncertainties?

Why it happens: Confusing rules. How to avoid it: ADD absolute uncertainties for SUMS/DIFFERENCES. ADD percentage uncertainties for PRODUCTS/QUOTIENTS. Source: 9702 Examiner Reports 2022-2024

Why is "Confusing precision with accuracy" a common mistake in Errors and uncertainties?

Why it happens: Similar in common speech. How to avoid it: Precision = consistency (low scatter). Accuracy = closeness to truth. A measurement can be precise but inaccurate. Source: 9702 Examiner Reports 2022-2024

Physical Quantities and UnitsCambridge International A Levels Physics (9702)

Errors and uncertainties

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Errors and Uncertainties Study Notes — Cambridge International A Level Physics 9702 (2025-2027 syllabus)

Random vs systematic errors. Absolute and percentage uncertainty. Propagation through addition, multiplication and powers.

What you’ll learn

Mapped to the Cambridge IGCSE 9702 syllabus (2025-2027).

1.3.1 — Distinguish random and systematic errors.
1.3.2 — Distinguish precision and accuracy.
1.3.3 — Calculate uncertainty in derived quantities.

Random vs systematic errors

Two error categories.

Random error. Unpredictable. Scatters readings about true value. Averaging multiple measurements REDUCES random error.

Systematic error. Consistent bias (all readings too high or too low). NOT reduced by averaging. Sources: calibration error, zero error, parallax, environment.

Precision. Repeated readings are close to each other (LOW random error).

Accuracy. Readings are close to TRUE value (LOW systematic error).

Can be either combination.

Precise + accurate: tight cluster around true value.
Precise + inaccurate: tight cluster but off-centre.
Imprecise + accurate: scattered but centred on true value.
Imprecise + inaccurate: scattered AND off-centre.

Cambridge tip. Mark scheme rewards both correct precision/accuracy AND error type identification.

Random: scatter, reduce by averaging.
Systematic: bias, averaging doesn't help.
Precision vs accuracy.

Expressing uncertainty

Absolute, fractional, percentage.

Absolute uncertainty. Uncertainty in the SAME UNITS as the measurement. e.g. $L = 4.20 \pm 0.05$ cm.

Fractional uncertainty. $\Delta x / x$ (dimensionless).

Percentage uncertainty. $\frac{\Delta x}{x} \times 100\%$ .

Significant figures convention. Uncertainty quoted to 1 sf; value to same decimal place.

e.g. $L = 4.20 \pm 0.05$ cm (NOT $L = 4.200 \pm 0.05$ ).

Example. $L = 4.20 \pm 0.05$ cm → percentage uncertainty $= \frac{0.05}{4.20} \times 100\% \approx 1.19\%$ .

Cambridge tip. Round uncertainty UP, not down. $1.2\%$ is acceptable; $1.0\%$ understates.

Absolute: same units.
Percentage: $\frac{\Delta x}{x} \times 100$ .
Uncertainty quoted to 1 sf.

See the full worked example for errors and uncertainties →

Propagation of uncertainty

Rules for combining.

Sum or difference: $z = x + y$ or $z = x - y$ . $\Delta z = \Delta x + \Delta y \quad (\text{add absolute uncertainties})$

Product or quotient: $z = xy$ or $z = x/y$ . $\%\Delta z = \%\Delta x + \%\Delta y$

Power: $z = x^n$ . $\%\Delta z = |n| \times \%\Delta x$

Method.

Identify the operations.
Apply matching rule.
Convert back to absolute if needed.

Example. Volume $V = \frac{4}{3}\pi r^3$ with $r = 5.0 \pm 0.1$ cm.

$\%\Delta r = 2\%$ .
$V \propto r^3$ → $\%\Delta V = 3 \times 2\% = 6\%$ .

Cambridge tip. When combining several uncertainties via different rules, work them out in stages.

Sum: add absolute.
Product: add percentage.
Power: multiply percentage by exponent.

See the full worked example for errors and uncertainties →

How it’s examined

Uncertainties appear every Paper 1 and Paper 3. Most-tested: percentage uncertainty (5 marks), propagation through formula (7-10 marks).

Sources: Cambridge International A Level Physics 9702 syllabus (2025-2027); 9702 Examiner Reports 2022-2024; 9702/12 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

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Step-by-step worked examples — Errors and uncertainties

Step-by-step solutions to past-paper-style questions on errors and uncertainties, written exactly the way a tutor would explain them at the board.

1Calculate percentage uncertainty (5 marks)
Extended• Adapted from 9702/12 May/Jun 2024• uncertainty
Question
A length is measured as $L = 4.20 \pm 0.05$ cm. Calculate the percentage uncertainty in $L$ . (5 marks)
Step-by-step solution
1. Step 1
  Percentage uncertainty = (absolute uncertainty / value) × 100%.
2. Step 2
  Substitute.
  $\%\Delta L = \dfrac{0.05}{4.20} \times 100\% \approx 1.19\%$
Answer
$\approx 1.2\%$ .
2Uncertainty propagation — product (7 marks)
Extended• propagation
Question
A rectangle has $L = 4.20 \pm 0.05$ cm and $W = 2.50 \pm 0.05$ cm. Calculate the area and its absolute uncertainty. (7 marks)
Step-by-step solution
1. Step 1
  Area $A = LW = 4.20 \times 2.50 = 10.50$ cm².
2. Step 2
  For products, percentage uncertainties ADD.
3. Step 3
  $\%\Delta L = 0.05/4.20 \times 100 \approx 1.19\%$ .
4. Step 4
  $\%\Delta W = 0.05/2.50 \times 100 = 2.00\%$ .
5. Step 5
  $\%\Delta A = 1.19 + 2.00 = 3.19\%$ .
6. Step 6
  Absolute uncertainty in area. $\Delta A = 10.50 \times 0.0319 \approx 0.34$ cm² (round to 1 sf).
Answer
$A = (10.5 \pm 0.3)$ cm².
3Uncertainty in a power (5 marks)
Extended• propagation
Question
The radius of a sphere is $r = 5.0 \pm 0.1$ cm. Calculate the percentage uncertainty in the volume $V = \frac{4}{3}\pi r^3$ . (5 marks)
Step-by-step solution
1. Step 1
  For powers, multiply percentage uncertainty by the power.
2. Step 2
  $\%\Delta r = 0.1/5.0 \times 100 = 2\%$ .
3. Step 3
  $V \propto r^3$ , so $\%\Delta V = 3 \times \%\Delta r = 6\%$ .
Answer
$\%\Delta V = 6\%$ .

Key Formulae — Errors and uncertainties

The formulae you need to memorise for errors and uncertainties on the Cambridge International A Level 9702 paper, with every variable defined in plain English and a note on when to use it.

Percentage uncertainty
$\%\Delta x = \dfrac{\Delta x}{x} \times 100\%$
$\Delta x$
absolute uncertainty
$x$
measured value
When to use
Express uncertainty as a percentage of the value.
Sum / difference
$z = x + y \text{ or } z = x - y \Rightarrow \Delta z = \Delta x + \Delta y$
When to use
Add ABSOLUTE uncertainties when adding or subtracting quantities.
Product / quotient
$z = xy \text{ or } z = x/y \Rightarrow \%\Delta z = \%\Delta x + \%\Delta y$
When to use
Add PERCENTAGE uncertainties when multiplying or dividing.
Powers
$z = x^n \Rightarrow \%\Delta z = |n| \times \%\Delta x$
When to use
Multiply percentage uncertainty by the (absolute) exponent.

Key Definitions and Keywords — Errors and uncertainties

Definitions to memorise and the exact keywords mark schemes credit for errors and uncertainties answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9702 sitting.

Random error
Examiner keyword
Unpredictable, scatters about true value. Reduced by averaging repeated readings.
Systematic error
Examiner keyword
Consistent in same direction. Not reduced by averaging. Caused by faulty equipment or method.
Precision
Examiner keyword
How close repeated readings are to each other. High precision = low random error.
Accuracy
Examiner keyword
How close measurement is to TRUE value. High accuracy = low systematic error.
Absolute uncertainty
Uncertainty expressed in same units as the measurement (e.g. $\pm 0.05$ cm).

Common Mistakes and Misconceptions — Errors and uncertainties

The traps other students keep falling into on errors and uncertainties questions — taken from recent Cambridge International A Level 9702 examiner reports and mark schemes — and how to avoid them.

✕Adding absolute uncertainties for products
9702 Examiner Reports 2022-2024
Why it happens
Confusing rules.
How to avoid it
ADD absolute uncertainties for SUMS/DIFFERENCES. ADD percentage uncertainties for PRODUCTS/QUOTIENTS.
✕Confusing precision with accuracy
9702 Examiner Reports 2022-2024
Why it happens
Similar in common speech.
How to avoid it
Precision = consistency (low scatter). Accuracy = closeness to truth. A measurement can be precise but inaccurate.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

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Errors and uncertainties — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Errors and uncertainties

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Calculate percentage uncertainty (5 marks)

2Uncertainty propagation — product (7 marks)

3Uncertainty in a power (5 marks)

Percentage uncertainty

Sum / difference

Product / quotient

Powers

Random error

Systematic error

Precision

Accuracy

Absolute uncertainty

✕Adding absolute uncertainties for products

✕Confusing precision with accuracy

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Errors and uncertainties — frequently asked questions