Pearson Edexcel International A Level Physics: Unit 3: processing results, graphs and uncertainties (YPH11) – Revision Notes

These notes condense section 3.5 Processing Results of Unit 3: Practical Skills in Physics I in the Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics (YPH11) specification, Issue 3. Unit 3 is an International AS (IAS) unit. For full explanations and longer worked examples, read the study guide. When you are ready to test yourself, use the practice questions, the Edexcel A Level Physics hub, the printable checklist and the free 10-minute diagnostics.

The specification states that students are not expected to compound percentage uncertainties. You only need the percentage uncertainty of a single measurement.

Key definitions

Term	Meaning
Resolution	The smallest change an instrument can show (smallest scale division, or last digit on a digital display)
Uncertainty	The doubt about a value, written as ± an amount
Random error	Readings scatter either side of the true value; repeating and averaging reduces it
Systematic error	Every reading is shifted the same way; repeating does not remove it
Precise	Repeat readings agree closely
Accurate	Close to the true value
Anomaly	A result that does not fit the trend; check it and repeat it
Zero error	A reading of non-zero when the quantity measured is actually zero

Formulae and rules

Quantity	Rule
Uncertainty, single reading	Half the resolution
Uncertainty, repeated readings	Half range = (largest − smallest) / 2
Percentage uncertainty	(uncertainty / value) × 100%
Gradient	(y₂ − y₁) / (x₂ − x₁), from two points on the best-fit line
Straight line	y = mx + c
Final answer	Same s.f. as the least precise data; carry extra digits in working

Method in steps: plotting and using a graph

1. Rearrange the equation into y = mx + c form, so you know what to plot and what the gradient means.
2. Label the axes with quantity and unit, and use a scale that fills more than half the grid, in easy steps.
3. Plot to half a small square and draw one best-fit line with a ruler, with points balanced either side.
4. Draw a large triangle on the line, using points far apart that are not data points.
5. Calculate the gradient with its unit, equate it to the expression, and rearrange.
6. Round the answer to the s.f. of the data.

Worked reminder: resistivity from a gradient

A wire of diameter 0.30 mm has resistance R measured for several lengths L. The best-fit line of R against L passes through (0.20 m, 1.46 Ω) and (1.00 m, 7.30 Ω).

Gradient = (7.30 − 1.46) / (1.00 − 0.20) = 5.84 / 0.80 = 7.3 Ω m⁻¹.

R = ρL / A, so the gradient is ρ / A and ρ = gradient × A.

A = π × (0.15 × 10⁻³)² = 7.07 × 10⁻⁸ m².

ρ = 7.3 × 7.07 × 10⁻⁸ = 5.2 × 10⁻⁷ Ω m (2 s.f., because the diameter has 2 s.f.).

What the shape of the graph tells you

Graph of y against x	Conclusion
Straight line through the origin	y is proportional to x
Straight line, non-zero intercept	Linear, not proportional; an intercept that should be zero suggests a systematic error
Curve	Not linear; plot against a transformed quantity such as x² or 1/x
Large scatter around the line	Large random error; repeat and average

Uncertainty: quick method

Single reading, for example a rule with 1 mm divisions reading 180 mm: uncertainty ±0.5 mm, percentage uncertainty 0.5 / 180 × 100 = 0.28%.

Repeated readings, for example 3.2 s, 3.4 s, 3.3 s: mean 3.3 s, half range 0.1 s, percentage uncertainty 0.1 / 3.3 × 100 = 3.0%.

Bigger measured quantities give smaller percentage uncertainties. That is why you time many oscillations, and why you measure a small diameter with a micrometer rather than a ruler.

Must-know distinctions

• Resolution is not uncertainty. For one reading, the uncertainty is half the resolution.
• Half range is not range. Divide by 2.
• Random error is cut by repeating. Systematic error is not, and needs a changed method or a corrected reading.
• Precision and accuracy are different. A biased instrument can be precise and still inaccurate.
• Improving the experiment is more than repeating it. Name the change and the error it fixes.

Realistic modifications

Problem	Modification
Reaction time when timing	Time many oscillations, or use a light gate and data logger
Starting and stopping at the wrong point	Fiducial marker at the centre of the swing
Ruler not vertical or parallax	Set square, plumb line, read at eye level
Zero error	Check the zero first, then correct every reading
Few readings or narrow range	More values over a wider range
Cannot judge the trend	Plot a graph rather than comparing pairs of readings
Wire may vary in thickness	Measure the diameter at several places and in perpendicular directions

Worked reminder: which measurement limits the result?

A student measures a spring extension of 24 mm with a rule of 1 mm divisions and a force with a newton meter of resolution 0.1 N reading 2.0 N.

Extension: 0.5 / 24 × 100 = 2.1%.

Force: 0.05 / 2.0 × 100 = 2.5%.

The force reading has the larger percentage uncertainty, so it is the measurement that limits the quality of the result. A sensible improvement is to use a larger load, so the same resolution gives a smaller percentage. You compare the two figures and stop there, as the specification does not expect you to combine them.

Writing a comment on a relationship

Use a short, three-part sentence: what the graph shows, what that means, and the evidence. For example: “The graph is a straight line through the origin, so the stopping distance is proportional to the square of the speed, because the plotted quantity is v² and the line passes through (0, 0) within the scatter.”

If the line misses the origin, say by how much and what it suggests, for example “an intercept of +0.03 m suggests a systematic error in measuring the starting height”.

Quick self-test

1. How many significant figures does 0.0460 have?
2. A voltmeter reads 2.4 V and its resolution is 0.1 V. Give the percentage uncertainty in the reading.
3. Three timings of a trolley are 3.2 s, 3.4 s and 3.3 s. Find the mean, then the percentage uncertainty using the half range.
4. A best-fit line passes through (0.10, 0.30) and (0.90, 2.70). What is its gradient?
5. A line that theory says must start at (0, 0) cuts the y-axis above zero. Which type of error does this point to?
6. Give one reason why the gradient triangle should be large.
7. Why does timing 20 swings give a smaller percentage uncertainty than timing 1?
8. A rule has 1 mm divisions and reads 18.0 cm. What is the percentage uncertainty?
9. A graph of R against L has gradient 12.4 Ω m⁻¹ for a wire of diameter 0.20 mm. Find the resistivity.
10. Calculate 2.4 × 3.15 to the correct number of significant figures when 2.4 has 2 s.f.

Answers

1. 3 s.f. The leading zeros do not count.
2. Uncertainty 0.05 V, so 0.05 / 2.4 × 100 = 2.1%.
3. Mean 3.3 s; half range (3.4 − 3.2) / 2 = 0.1 s; 0.1 / 3.3 × 100 = 3.0%.
4. (2.70 − 0.30) / (0.90 − 0.10) = 3.0.
5. A systematic error, for example a zero error.
6. Reading error in each coordinate is a smaller fraction of the large change in x and y, so the gradient is more reliable.
7. The same timing uncertainty, from reaction time, is divided by a much longer total time.
8. Uncertainty 0.5 mm on 180 mm gives 0.28%.
9. A = π × (0.10 × 10⁻³)² = 3.14 × 10⁻⁸ m²; ρ = 12.4 × 3.14 × 10⁻⁸ = 3.9 × 10⁻⁷ Ω m.
10. 7.56 on the calculator, so 7.6 to 2 s.f.

Where marks are usually lost

• Writing every digit on the calculator display as the final answer.
• Plotting with awkward scales, or with a graph that fills less than half the grid.
• Leaving off axis units, or writing a unit that does not match the quantity.
• Drawing a gradient triangle that is small, or using data points that are not on the line.
• Forgetting to convert the gradient into the quantity asked for, such as halving or multiplying by an area.
• Using the full range, or the full resolution, as the uncertainty.
• Suggesting “use more accurate equipment” without saying which instrument or what it improves.
• Describing a repeat as a cure for a systematic error.
• Saying a curve is proportional because it rises.

Official syllabus

Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics (YPH11), Specification, Issue 3, published by Pearson Education Limited. Section 3.5 Processing Results, within Unit 3: Practical Skills in Physics I.