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

This guide is aligned to the Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (YPH11), Issue 3, and covers section 3.5 Processing Results in Unit 3: Practical Skills in Physics I. Unit 3 is an International AS (IAS) unit. In this part of the unit you are given results collected by a more-experienced student and you process them. The specification says that compounding percentage uncertainties is not expected, so none is used here.

Related pages: the revision notes, the practice questions, the Edexcel A Level Physics hub, the printable checklist and the free 10-minute diagnostics. The graph skills here are used on content from Materials and Electric circuits.

What section 3.5 asks you to do

Specification statement	In practice
Calculations with the correct number of significant figures	Round the final answer sensibly, keep spare digits in the working
Plot a graph with an appropriate scale	Use most of the paper, simple scales, labelled axes with units
Use the correct units throughout	Column headings, axis labels, gradients, final answers
Comment on the relationship from the graph	Straight line through the origin, straight line with an intercept, curve
Determine a relationship or a constant, for example a gradient with a large triangle	Turn the gradient into the physical quantity asked for
Realistic modifications that cut errors or improve the method	Specific, named changes that fix a named problem
Discuss uncertainties qualitatively and quantitatively	Random against systematic; percentage uncertainty
Percentage uncertainty from half the resolution, or from the half range	Two methods, covered below

The specification gives the assessment of this unit as 1 hour and 20 minutes and says that calculators may be used in the examination.

Significant figures in calculations

The number of significant figures (s.f.) in a result should reflect the data used to get it. A sensible working rule is to give the final answer to the same number of s.f. as the least precise value used, and to carry at least one extra digit through the working so that rounding does not build up.

Zeros that only place the decimal point do not count: 0.0046 has 2 s.f. A trailing zero after the decimal point does count: 0.250 has 3 s.f. Say so when a quantity is 2 s.f. and the calculator shows many more digits.

Raw readings should be recorded to the resolution of the instrument. A micrometer reading 0.46 mm should not be recorded as 0.5 mm or as 0.4600 mm.

Worked example 1: density of a cylinder

A cylinder has mass 52.3 g, diameter 2.46 cm and length 7.35 cm. All three values have 3 s.f.

Volume = π × d² / 4 × length = π × (2.46)² / 4 × 7.35 = 34.93 cm³ (keeping extra digits).

Density = 52.3 / 34.93 = 1.497 g cm⁻³.

Final answer: 1.50 g cm⁻³, or 1.50 × 10³ kg m⁻³, both 3 s.f. Writing 1.497112 would claim a precision the readings do not have.

Plotting a graph

1. Choose the axes. The independent variable (the one you changed) goes on the x-axis. If the question names the axes, follow it.
2. Label each axis with the quantity and its unit, for example v² / m² s⁻².
3. Choose a scale so that the points fill more than half of the grid in both directions. Use steps that are easy to read, such as 1, 2 or 5 small squares per unit. Avoid awkward steps such as 3 squares for 1 unit.
4. Include the origin only if it helps. If you need the intercept, or the line is expected to go through the origin, include it.
5. Plot each point to within half a small square, using a small neat cross or dot.
6. Draw one straight line of best fit with a ruler, so that points lie evenly on both sides. If the points curve, draw a smooth curve instead. Do not join the dots.

An isolated point far from the trend is an anomaly. Check it: if the calculation or plotting is wrong, correct it. If it is a genuine reading, say that it should be repeated and do not drag the line towards it.

Reading the relationship from a graph

What the graph shows	What it means
Straight line through the origin	y is directly proportional to x
Straight line with a positive or negative intercept	y = mx + c; the relationship is linear but not proportional
Curve	Not linear; try plotting y against x², 1/x or √x

Many experiments are designed so that a derived quantity gives a straight line. If the theory says s = v²/(2a), then plotting v² against s gives a straight line through the origin with gradient 2a.

An intercept that should be zero, but is not, is evidence of a systematic error, for example a zero error on an instrument.

Gradients and large triangles

The gradient is the change in y divided by the change in x, taken from the line of best fit and not from data points.

1. Pick two points on the line that are far apart, using more than half the length of the line.
2. Choose points that sit on grid crossings so they can be read accurately.
3. Subtract: gradient = (y₂ − y₁) / (x₂ − x₁), keeping the axis units.
4. Equate the gradient to the physical expression and rearrange.

A large triangle is used because a reading error of half a small square matters much less over a long baseline.

Worked example 2: acceleration of a trolley

A trolley runs down a ramp from rest. For several distances s, a light gate gives the speed v at the end of the run.

s / m	0.20	0.40	0.60	0.80	1.00	1.20
v / m s⁻¹	0.90	1.26	1.53	1.79	1.99	2.18
v² / m² s⁻²	0.810	1.59	2.34	3.20	3.96	4.75

Since v² = 2as, a graph of v² against s is a straight line through the origin with gradient 2a.

Scale: with 12 large squares up the page, 0.5 m² s⁻² per large square covers 0 to 6.0 and fills the grid. Along the page, 0.1 m per large square covers 0 to 1.2 m.

The best-fit line passes through (0.20, 0.80) and (1.20, 4.76).

Gradient = (4.76 − 0.80) / (1.20 − 0.20) = 3.96 / 1.00 = 3.96 m s⁻².

a = gradient / 2 = 1.98 m s⁻².

Check the units: m² s⁻² divided by m gives m s⁻², the unit of acceleration. The line passes through the origin within the scatter, so v² is proportional to s, as the theory predicts.

Uncertainties

Qualitative ideas

Uncertainty is the doubt about a measured value. A random error makes readings scatter either side of the true value, so repeating and averaging reduces its effect. A systematic error shifts every reading the same way, for example a zero error, and averaging does not remove it. It shows up on a graph as an intercept that should be zero, or as a line parallel to the expected one.

A result is precise if repeats agree closely. It is accurate if it is close to the true value. Precise readings can still be inaccurate if there is a systematic error.

Common sources are parallax when reading scales, reaction time when starting and stopping a stopwatch, a ruler that is not vertical or not touching the object, and an instrument that does not read zero when it should.

Percentage uncertainty from one reading

For a single reading the uncertainty is half the resolution of the instrument.

Percentage uncertainty = (uncertainty / reading) × 100%.

Example: a metre rule with 1 mm divisions gives an uncertainty of ±0.5 mm. For a length of 250 mm this is 0.5 / 250 × 100 = 0.2%.

For a micrometer with resolution 0.01 mm, the uncertainty is ±0.005 mm. On a wire of diameter 0.28 mm this is 0.005 / 0.28 × 100 = 1.8%.

The same instrument gives a much larger percentage uncertainty on a small quantity than on a large one. That is why thin wires are measured with a micrometer and not a ruler.

Percentage uncertainty from repeated readings

For several repeats, the uncertainty is the half range: (largest − smallest) / 2. Use the mean as the reading.

Worked example 3: wire diameter

Five micrometer readings of a wire, in mm: 0.31, 0.32, 0.31, 0.33, 0.31.

Mean = 1.58 / 5 = 0.316 mm.

Half range = (0.33 − 0.31) / 2 = 0.010 mm.

Percentage uncertainty = 0.010 / 0.316 × 100 = 3.2%.

Quote the diameter as 0.32 ± 0.01 mm.

The specification does not expect you to combine percentage uncertainties from several measurements, so you stop at this stage. If the question asks which measurement has the larger percentage uncertainty, compare them and give the reason.

Reducing errors and improving the experiment

Good suggestions are specific and linked to a named problem. Compare these.

Weak	Better
“Be more careful”	“Time 20 oscillations and divide by 20, so that the reaction time of about 0.2 s is a smaller fraction of the total”
“Use better equipment”	“Use a light gate connected to a data logger, which removes the reaction time”
“Repeat it”	“Repeat each reading three times and take the mean, to reduce random error”

Worked example 4: why timing more oscillations helps

Take a pendulum with a period of 1.4 s and an assumed reaction-time uncertainty of 0.2 s.

Timing one oscillation: 0.2 / 1.4 × 100 = 14%.

Timing 20 oscillations gives a total of 28.0 s: 0.2 / 28.0 × 100 = 0.71%.

The period is 28.0 / 20 = 1.40 s. The percentage uncertainty is cut by a factor of 20 because the same absolute timing uncertainty now applies to a longer interval.

Other realistic modifications:

• Use a set square or plumb line to check a ruler or stand is vertical.
• Use a fiducial marker at the centre of the swing, so the count starts and stops at a fixed point.
• Take more readings over a wider range, then plot the graph, so a trend is clearer than from a single point.
• Zero an instrument first, or record the zero error and correct every reading.
• Use a larger load or length so the quantity measured is bigger and its percentage uncertainty is smaller.

Common errors

• Giving a final answer to 6 s.f. just because the calculator does.
• Plotting on a cramped scale, so the line fills a quarter of the grid.
• Drawing the gradient triangle with a short baseline, or using data points not on the line.
• Leaving out the unit of a gradient, or forgetting to halve or double it when the equation says so.
• Calling a scatter of points a systematic error, or an offset a random error.
• Using the full range, not the half range, as the uncertainty.
• Using the resolution as the uncertainty for a single reading instead of half the resolution.

Finish with the practice questions, then use the revision notes for quick recall.

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.