In simple terms
A friendly intro before the formal notes — no formulas yet.
Errors and Uncertainties in Paper 5
Master errors and uncertainties for Cambridge A-Level Physics Paper 5. Learn to distinguish systematic and random errors, calculate uncertainties in derived quantities, and apply these skills to graphical analysis and experimental planning. Syllabus coverage: understand and explain the effects of systematic errors (including zero errors) and random errors in measurements understand the distinction between precision and accuracy assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties
- 1
Distinguish between systematic and random errors and their effects on experimental data.
- 2
Explain the difference between accuracy and precision.
- 3
Calculate absolute and percentage uncertainties in derived quantities.
- 4
Determine the uncertainty in the gradient and y-intercept of a line of best fit.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 1.3.1
Understand and explain the effects of systematic errors (including zero errors) and random errors in measurements
- 1.3.2
Understand the distinction between precision and accuracy
- 1.3.3
Assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
Distinguish between systematic and random errors and their effects on experimental data.
5 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
5 simulations · 4 to start with
Start herein this order — each one shows a different piece of the topic
- PhETStart here · 19702 Paper 5
Least-Squares Regression
Add points, draw your own line, then compare it with the least-squares best fit.
Why this one: Draw your own best line, then compare its gradient and intercept with the least-squares fit.
Try this
- Add 8 points and draw the line you think is best.
- Reveal the best-fit line — compare your gradient and intercept with the sim’s.
- Drag one point vertically and watch which moves more: gradient or intercept.
Look for The best-fit line balances the squared vertical distances — worst and best lines bracket the true gradient.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- PhETStart here · 29702 2.1 · 1.3 · Paper 5 · IB A.1
Projectile Data Lab
Fire many projectiles from one launcher and analyse the landing positions with histograms, the mean and the spread.
Why this one: Fire the same shot 20 times: the scatter is random error, the mean settles as readings grow.
Try this
- Fire 20 shots at the same setting — read the spread of landing distances.
- Open the Measures screen — compare the mean with any single shot.
- Fire 100 more — watch the standard error of the mean shrink.
Look for Repeated readings scatter; the mean settles and its uncertainty falls as more readings are taken.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- PhETStart here · 39702 1.3
Plinko Probability
Drop balls through a peg board and watch the bins build a binomial histogram that tends to a normal curve.
Why this one: Many small random nudges pile up into a bell curve, which is why repeats cluster around a mean.
Try this
- Drop balls one at a time — note how scattered the first ten land.
- Switch to continuous drop — the histogram smooths into a bell shape.
- Change the number of rows — the width of the bell changes.
Look for Many small random deviations add to a normal distribution: why repeated readings cluster around a mean.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- 3JCN PhysicsStart here · 49702 18.3 · 9702 1.3 · 9702 Paper 5
Coulomb's Law for Lab
Coulomb's law for lab: collect force-distance data and fit the inverse square
Why this one: Collect force readings, plot F against 1/r² and judge the straight line like a Paper 5 analysis.
Try this
- Collect force readings at several separations.
- Plot force against 1/r² and check for a straight line.
Look for Force against 1/r² is a straight line through the origin, confirming the inverse square law.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
More simulations1 more on this topic — core ones first
- 3JCN PhysicsCore9702 17.1 · 9702 17.2 · 9702 1.3
SHM for Lab
Timed mass-spring oscillator for lab: change mass and k and measure the period
Try this
- Set a mass and k, time ten oscillations and divide for the period.
- Double the mass and measure the period again.
- Double k instead and measure the period again.
Look for T = 2π√(m/k): quadrupling the mass doubles the period.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
Full topic notes
Formal explanation with the rigour you need for the exam.
Understanding Experimental Errors
Every measurement you take is subject to errors. These are not 'mistakes' but inherent variations in the measurement process. They are broadly classified into two types: systematic errors and random errors.
Systematic Errors: These are consistent, repeatable errors that cause your measurements to be consistently off from the true value in the same direction. For example, a voltmeter that has not been zeroed correctly (a zero error) will give every reading as slightly too high or too low. Systematic errors affect the accuracy of your results. They cannot be reduced by repeating measurements and averaging.
Random Errors: These are unpredictable variations in measurements. They can be caused by factors like fluctuations in environmental conditions, misreading a scale, or variations in reaction time. Random errors cause a scatter of readings around a mean value. They affect the precision of your results. The effect of random errors can be reduced by taking multiple readings and calculating an average.
Accuracy vs. Precision
These two terms are often used interchangeably in everyday language, but in physics, they have distinct meanings.
- Accuracy is how close a measured value is to the true or accepted value.
- Precision is how close repeated measurements are to each other.
Imagine an archer shooting at a target:
- Accurate and Precise: All arrows are in the bullseye.
- Precise but not Accurate: All arrows are clustered together, but far from the bullseye. This suggests a systematic error (e.g., the bow's sight is misaligned).
- Accurate but not Precise: The arrows are scattered all over the target, but their average position is the bullseye. This suggests significant random errors.
- Neither Accurate nor Precise: The arrows are scattered all over the target, and their average is not in the bullseye.
In Paper 5 Question 1 (Planning), when asked to evaluate or suggest improvements to an experiment, consider sources of systematic and random errors. For example, suggesting the use of a more precise instrument (e.g., a micrometer instead of a ruler) addresses random error, while suggesting a method to check for a zero error addresses a systematic error.
Calculating and Propagating Uncertainties
When you use measured quantities to calculate a new quantity, the uncertainties in your measurements 'propagate' to the final result. You need to know the rules for combining them.
In Paper 5, you'll often need to calculate uncertainties for values in a table, especially for quantities that are reciprocals or logarithms of your measurements.
For a reciprocal, : The fractional uncertainty rule gives . Rearranging for the absolute uncertainty gives .
For a logarithm, : The simplest method is to find the range: . Alternatively, you can use . Both are acceptable.
Uncertainties in Graphical Analysis
A core task in Paper 5 is to determine the gradient and y-intercept of a graph and find the uncertainties in these values. This is done by drawing a line of best fit (LOBF) and a worst acceptable line (WAL).
- Plot error bars: For each point, draw a vertical bar representing its absolute uncertainty.
- Draw the LOBF: Draw a single straight line that passes as close as possible to all data points, with a balance of points above and below the line.
- Draw the WAL: This is the line with the steepest or shallowest gradient that still passes through all the error bars. You only need to draw one WAL (either steepest or shallowest). To draw the steepest WAL, pivot the line around the lowest error bar until it touches the highest error bar. For the shallowest, pivot around the highest error bar until it touches the lowest.
Systematic errors affect accuracy; random errors affect precision.
Always add absolute uncertainties for addition/subtraction.
Always add percentage/fractional uncertainties for multiplication/division/powers.
Uncertainty in a gradient or intercept is found by taking the difference between the best fit and worst acceptable values.
A worst acceptable line is the steepest or shallowest line that passes through all error bars.
Always use a 'large triangle' when calculating a gradient from a graph. This means the points you choose on the line should be separated by at least half the length of the drawn line. This minimises the uncertainty in your calculation.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
Fig. 1.1 shows a thin cylindrical metal rod of length .
[Figure: Fig. 1.1 shows a thin cylindrical metal rod of length L.]
One end of the rod is hit with a hammer. A stationary sound wave is set up within the rod. The rod vibrates at its resonant frequency .
A microphone placed at the other end of the rod detects the sound wave emitted from the rod. The frequency of the detected sound is also .
A number of rods of different length are available.
It is suggested that is related to by the relationship where is the density of the metal, and and are constants.
Plan a laboratory experiment to test the relationship between and .
Draw a diagram showing the arrangement of your equipment.
Explain how the results could be used to determine values for and .
In your plan you should include:
- the procedure to be followed
- the measurements to be taken
- the control of variables
- the analysis of the data
- any safety precautions to be taken.
- 1
is the independent variable and is the dependent variable, or vary and measure .
Values of and are given in Table 2.1.
| 1.25 | ||
| 2.55 | ||
| 3.90 | ||
| 5.25 | ||
| 6.55 | ||
| 7.80 |
Calculate and record values of in Table 2.1.
Include the absolute uncertainties in .
| 1.25 | ||
| 2.55 | ||
| 3.90 | ||
| 5.25 | ||
| 6.55 | ||
| 7.80 |
- 1
Correctly calculated values for in the table.
Values of and are given in Table 2.1.
Table 2.1
| 54 | |||
| 70 | |||
| 86 | |||
| 108 | |||
| 140 | |||
| 167 |
Calculate and record values of and in Table 2.1. Include the absolute uncertainties in .
| 54 | |||
| 70 | |||
| 86 | |||
| 108 | |||
| 140 | |||
| 167 |
- 1
Values of lg ( / cm) and lg ( s) correct as shown above.
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- Accuracy
Accuracy is how close a measured value is to the true or accepted value.
- Precision
Precision is how close repeated measurements are to each other.
- Systematic error
Consistent, repeatable error in the same direction — affects accuracy, not fixed by averaging.
- Random error
Unpredictable scatter between readings — affects precision; reduced by repeating and averaging.
- Accuracy vs precision
Accuracy = closeness to true value; precision = closeness of repeated readings to each other.
- Adding absolute
When adding or subtracting quantities, add absolute uncertainties: .
- Multiplying percentage
When multiplying/dividing, add fractional uncertainties: .
- Uncertainty in gradient
Gradient uncertainty = |gradient of best fit ? gradient of worst acceptable line|.
- Worst acceptable line
Steepest or shallowest straight line still passing through all error bars on the graph.
- Zero error example
A voltmeter not zeroed — every reading shifted by the same amount (systematic).
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
Systematic Errors: These are consistent, repeatable errors that cause your measurements to be consistently off from the true value in the same direction. For example, a voltmeter that has not been zeroed correctly (a zero error) will give every reading as slightly too high or too low. Systematic errors affect the accuracy of your results. They cannot be reduced by repeating measurements and averaging.
Random Errors: These are unpredictable variations in measurements. They can be caused by factors like fluctuations in environmental conditions, misreading a scale, or variations in reaction time. Random errors cause a scatter of readings around a mean value. They affect the precision of your results. The effect of random errors can be reduced by taking multiple readings and calculating an average.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Fig. 1.1 shows a thin cylindrical metal rod of length . [Figure: Fig. 1.1 shows a thin cylindrical metal rod of length L.] One end of the rod is hit with a hammer. A stationary sound wave is set up within the rod. The rod vibrates at its resonant frequency . …
Values of and are given in Table 2.1. Table 2.1 | | | | | …
An electric pump is placed in a container of liquid. A model wind turbine is connected to the pump by a cable, as shown in Fig. 1.1. [Figure: Fig. 1.1 shows a model wind turbine with blades connected by a cable to a pump submerged in a container of liquid. The pump pushes liquid up a vertical pipe to a height h. Moving air causes the turbine to rotate.] The turbine is placed in moving air. As the turbine blades turn, electricity is generated and the pump pushes liquid through a vertical pipe. …
Values of and are given in Table 2.1.
| … |
Plot a graph of against . Include error bars for .
Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines.
Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer.
Determine the y-intercept of the line of best fit. Include the absolute uncertainty in your answer.
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do 52 · February/March 2024 · Q1 on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
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