Pearson Edexcel International A Level Physics: Unit 6: analysing data, log graphs and uncertainties (YPH11) – Revision Notes

These revision notes condense section 6.5 Analysis of Unit 6: Practical Skills in Physics II in the Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (YPH11), Issue 3. Unit 6 is part of IA2 and has no tiers. For full explanations and longer worked examples, use the study guide. Then test yourself with the practice questions, check coverage on the printable checklist, see the course hub and take a free 10-minute diagnostic.

Key terms

Term	Meaning in this specification
Accuracy	Closeness of a result to the true value; cannot be quantified; affected by random and systematic errors
Precision	Closeness of agreement between repeated measurements; affected only by random effects
Repeatability	Precision when one operator, one method and a short timescale are used
Reproducibility	Precision when different operators or apparatus are used; a harder test
Resolution	Smallest measuring interval; the source of uncertainty in a single reading
Uncertainty	The interval within which the true value can be taken to lie, such as 20 °C ± 2 °C
Error	Difference between a result and the true value; not a mistake
Validity	A measurement is valid if it measures what it is meant to measure
Sensitivity	Not defined in the specification glossary. Use it for how much an instrument’s reading changes for a small change in the quantity

Systematic errors shift every reading the same way and can be corrected if known, for example by subtracting a zero error. Random errors scatter the readings, and repeating and averaging reduces their effect.

Formulas and rules

Task	Rule
Percentage uncertainty	(uncertainty / measurement) × 100%
Single reading	Uncertainty = half the resolution
Repeated readings	Uncertainty in the mean = half the range
Multiply or divide	Add percentage uncertainties
Raise to a power n	Percentage uncertainty is n times that of the quantity
Add or subtract	Add the absolute uncertainties
Percentage difference	(measured - accepted) / accepted × 100%
Power law y = kxⁿ	log y = n log x + log k: gradient n, intercept log k
Exponential y = Ae^(-bx)	ln y = ln A - bx: gradient -b, intercept ln A

Method in steps: uncertainty in a result

1. Write each measured value with its absolute uncertainty.
2. Convert each to a percentage uncertainty, to 1 or 2 significant figures.
3. Count powers: a squared quantity counts twice, a cubed quantity three times.
4. Add the percentages for products and quotients.
5. Convert back to an absolute uncertainty: result × total % / 100.
6. Round the uncertainty to 1 or 2 significant figures and match the result to it.

Worked reminder. A trolley of mass 0.240 kg ± 0.002 kg moves at 3.40 m s⁻¹ ± 0.05 m s⁻¹. Find its kinetic energy and the uncertainty.

KE = ½mv² = 0.5 × 0.240 × 3.40² = 1.387 J, which is 1.39 J
% in m = 0.002 / 0.240 × 100 = 0.83%
% in v = 0.05 / 3.40 × 100 = 1.5%
total % = 0.83% + 2 × 1.5% = 3.8%
absolute = 1.39 × 3.8 / 100 = 0.05 J

Result: KE = 1.39 J ± 0.05 J (3.8%).

Subtraction reminder. Lengths 15.0 cm ± 0.2 cm and 8.5 cm ± 0.1 cm added give 23.5 cm ± 0.3 cm. Absolute uncertainties add for both sums and differences.

Method in steps: a straight-line graph from a log plot

1. Decide the form: a power law needs log y against log x; an exponential needs ln y (or log y) against x.
2. Work out the logged column from the numbers only, never the units, and keep 3 significant figures or a consistent number of decimal places.
3. Label axes as log (quantity/unit) or ln (quantity/unit).
4. Plot, using more than half of each axis, and draw a thin best-fit line.
5. Choose two points on the line far apart. Read them carefully and compute rise ÷ run.
6. Link the gradient to the physics: n for a power law, -b or -1/τ for a decay.
7. Read the intercept and convert it back: k = 10^intercept for a log₁₀ plot, A = e^intercept for an ln plot.

Straight and passing through the expected point means the model is supported. Curved means the assumed relationship is wrong. Using log₁₀ in place of ln for an exponential multiplies the gradient by 0.434.

Must-know distinctions

• Precision (spread) is separate from accuracy (closeness to true value). Tight readings can still be wrong because of a systematic error.
• Resolution is the smallest division. The uncertainty in one reading is half of it.
• Percentage uncertainty of a radius equals that of the diameter. Powers apply only after you have the percentage.
• Absolute uncertainties are added for sums and differences. Percentages are added for products and quotients.
• A line of best fit that misses the origin when it should not indicates a systematic error. Large scatter points to random error.
• Accuracy against a known value: use the range from the percentage uncertainty when you have it. Use the percentage difference only when the uncertainty is not known, and treat under 5% as accurate.
• Error bars are not expected on graphs.

Improvements and modifications: stock ideas

Problem	Realistic change
Reaction time on short times	Time many oscillations, or use a light gate or data logger
Large percentage uncertainty in a small length	Use a micrometer or vernier caliper, or measure a longer length
Rule not vertical or level	Use a set square or spirit level
Zero error	Check zero first; subtract the error
Few readings or narrow range	Take more readings over a wider range, and repeat each
Heating of a wire or component	Use low current, switch off between readings
Drift in a variable	Control it with a thermostat bath, or wait for it to settle

Reporting results

• State the value, the uncertainty and the unit together, for example 1.39 J ± 0.05 J, with the uncertainty to 1 or 2 significant figures.
• A total percentage uncertainty under about 5% suggests the measurement is repeatable.
• To compare with a known value, work out the range the result allows. If the known value falls inside it, say the result is accurate within the uncertainty.
• Give each processed column of a table a consistent number of significant figures, usually 3. Mixing 3 and 4 is acceptable only when the data cross a power of ten.
• Say what the percentage uncertainty is dominated by and name the one measurement to improve.

Quick self-test

1. A rule has a resolution of 1 mm and gives 37 mm for a single length. Find the percentage uncertainty.
2. Three timings are 12.1 s, 12.6 s and 12.3 s. Give the mean and the percentage uncertainty from the half range.
3. Speed = distance ÷ time, with the distance ±1.2% and the time ±3.0%. Find the percentage uncertainty in the speed.
4. Power P = I²R, with the current ±2% and the resistance ±3%. Find the percentage uncertainty in P.
5. Two lengths, 15.0 cm ± 0.2 cm and 8.5 cm ± 0.1 cm, are added. What is the total and its absolute uncertainty?
6. A log₁₀ y against log₁₀ x graph is a straight line of gradient 1.5 and passes through (0, 0.48). Write y as a function of x.
7. ln N against t has a gradient of -0.35 min⁻¹. Find the half-life.
8. Give 6.1 × 2.35 to the correct number of significant figures.
9. A measured value of 9.5 m s⁻² is compared with an accepted 9.81 m s⁻². Find the percentage difference and decide if it is accurate by the 5% guide.
10. A graph that should pass through the origin has a positive intercept on the y-axis. What does this suggest, and what could you do?

Answers

1. Uncertainty 0.5 mm; 0.5 / 37 × 100 = 1.35%, so 1.4%.
2. Mean = 12.33 s, half range = 0.25 s; 0.25 / 12.33 × 100 = 2.0% (mean 12.3 s).
3. Add the percentages: 1.2% + 3.0% = 4.2%.
4. 2 × 2% + 3% = 7%.
5. Total 23.5 cm ± 0.3 cm.
6. Gradient 1.5 gives n = 1.5. The intercept 0.48 means k = 10^0.48 = 3.0, so y = 3.0 x^1.5.
7. λ = 0.35 min⁻¹, half-life = ln 2 / 0.35 = 2.0 min.
8. The least precise value has 2 significant figures: 14.335 becomes 14.
9. (9.81 - 9.5) / 9.81 × 100 = 3.2%, below 5%, so the result is accurate by this guide.
10. A systematic error, such as a zero error or a constant offset. Check the instrument zero, correct each reading, and repeat.

Where marks are usually lost

• Writing an answer with far more significant figures than the data justify.
• Leaving units out, or leaving a prefix unconverted before substituting.
• Using the full resolution as the uncertainty of a single reading.
• Adding percentages for a sum or difference, or adding absolute values for a product.
• Not multiplying by the power when a quantity is squared or cubed.
• Drawing a gradient triangle that is small or uses data points rather than points on the line.
• Using the words precise and accurate as if they were the same.
• Giving vague improvements such as “be more careful” or “use better equipment”, with no named instrument, variable or effect.
• Stating a conclusion about a law with no reference to the straight line, the gradient or the scatter.

Official syllabus

Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (YPH11), Issue 3, Pearson Education Limited, section 6.5 Analysis and the glossary in Appendix 10.