Key Idea: This topic is the experimental-skills toolkit — how you take a measurement, attach an uncertainty to it, push that uncertainty through a calculation, and turn a table of readings into a straight-line graph you can read a physics quantity off.
Measure
- Resolution suits the size
- Measure a multiple, then divide
- Change one, measure one, control the rest
Uncertainty
- ± half the range of repeats
- × ÷ add fractions; ± add absolutes
- Uncertainty to 1 s.f.
Graph and judge
- Gradient off the line; Δm = (max − min)/2
- Linearize: powers, 1/x, ln, log–log
- Intercept where none expected: systematic
These skills are examined as a whole paper of their own — Paper 1B (data analysis) — built around one experiment. Expect: pick the right instrument and justify it; quote and combine uncertainties; draw a best-fit line and read its gradient (with the steepest/shallowest lines for the gradient's uncertainty); linearize a law and decide whether the data support it; and evaluate the method (random vs systematic error, repeats, anomalies). The same habits — units, significant figures, ± uncertainties — also score easy marks all through Paper 2.
Key rules & formulas
The multiply/divide and power propagation rules carry the data-booklet badge (look for it). The add/subtract rule, the gradient and the gradient-uncertainty are not printed — you reproduce those from the definitions.
- the calculated result (e.g. a density = m ÷ V)
- the measured quantities multiplied or divided to get y
- the fractional uncertainty in a (no unit)
- the fractional uncertainty in the result y
- the result is a raised to a power n (e.g. area = πr², so r²)
- the power (2 for a square, 3 for a cube, ½ for a square root)
Not in the booklet: know these from the definitions.
- absolute uncertainty (same unit as x)
- gradient and intercept of the best-fit line
- the constant and the power found from a log graph
Choosing the instrument (resolution)
| Instrument | Resolution (smallest division) | Use it to measure |
|---|---|---|
| Metre rule | 1 mm | lengths from a few cm up to about 1 m |
| Vernier caliper | 0.1 mm | the diameter of a marble or the width of a block |
| Micrometer screw gauge | 0.01 mm | the thickness of a wire or a single sheet of paper |
| Measuring cylinder | ≈ 1 mL | the volume of a liquid |
| Stopwatch | 0.01 s | a time interval (time several swings, then divide) |
Choose the instrument whose smallest division is small compared with what you are measuring. A 0.5 mm wire on a mm ruler is hopeless (±0.5 mm is the whole thing) but easy on a micrometer (±0.005 mm). Reading uncertainty from a single instrument = ± half its smallest division.
The three propagation rules side by side
| Operation | What you do with the uncertainties | Booklet? |
|---|---|---|
| Add or subtract (+ , −) | Add the ABSOLUTE uncertainties: Δy = Δa + Δb | No — derived |
| Multiply or divide (× , ÷) | Add the FRACTIONAL (or %) uncertainties: Δy/y = Δa/a + Δb/b + … | Given (6.1) |
| Power (aⁿ) | Multiply the fractional uncertainty by |n|: Δy/y = |n|·Δa/a | Given (6.1) |
Random vs systematic error
| Random error | Systematic error | |
|---|---|---|
| What it does | Scatters readings either side of the true value | Shifts EVERY reading the same way (a zero error, a wrong calibration) |
| Affects | Precision (the spread) | Accuracy (how close to the true value) |
| Repeat & average? | Yes — the scatter partly cancels, shrinking it | No — averaging does nothing; you must fix the instrument or method |
| On a graph | Points scatter about the best-fit line | A non-zero intercept, or every point shifted |
A straight line alone only means the law is linear (Y = mX + c). It is directly proportional only if the line also passes through the origin (c = 0). Equivalently, the ratio Y/X is constant across every row of the table.
Worked exam-style questions
A student measures the thickness of one sheet of paper. A micrometer (resolution 0.01 mm, so ±0.005 mm) on a single sheet would give a fractional uncertainty of nearly 7%. Instead they measure a stack of 80 sheets and get 8.40 mm with the same ±0.005 mm. (a) Find the thickness of one sheet. (b) Find the absolute and percentage uncertainty in the thickness of one sheet, and say why this method is better.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
A small cube has side L = 2.00 ± 0.01 cm and mass m = 64.0 ± 0.5 g. The density is ρ = m ÷ V, where V = L³. (a) Find the density. (b) Find its percentage uncertainty. (c) Quote ρ with its absolute uncertainty.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
On a force-versus-extension graph the best-fit line passes through (0, 0). Two points read off THE LINE are (0.020 m, 2.4 N) and (0.080 m, 9.6 N). The steepest and shallowest lines through the error bars have gradients 124 N m⁻¹ and 116 N m⁻¹. (a) Find the spring constant from the best-fit gradient. (b) Find the uncertainty in the gradient and quote k properly.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Next:
Theory predicts the depth d a marker sinks is d = k√P, where P is the water pressure. (a) State what to plot on each axis to get a straight line through the origin, and what the gradient represents. (b) A classmate instead claims d is directly proportional to P. Using the rows (P = 4.0 kPa, d = 3.1 cm) and (P = 9.0 kPa, d = 4.6 cm), show that this claim is wrong.
Model answer plan
See the mark-by-mark plan — for / against / judgement, with marking guidance — in study mode.
Important: Use two far-apart points on the best-fit line, never two plotted readings. And a straight line is only proportional if it also passes through the origin: an intercept where none is expected points to a systematic error.
Quick self-check
Tap each card to reveal the answer.
Which instrument for a 0.5 mm wire diameter, and why? A micrometer (resolution 0.01 mm). Its smallest division is tiny compared with 0.5 mm, so the percentage uncertainty is small. A mm ruler (±0.5 mm) would be useless here.
What are the three ways to quote an uncertainty? Absolute (a ± value in the unit), fractional (absolute ÷ value), and percentage (fractional × 100%). They all carry the same information.
How do uncertainties combine for × and ÷, and for a power aⁿ? For × / ÷ add the fractional uncertainties (Δy/y = Δa/a + Δb/b + …). For a power multiply the fractional uncertainty by |n|. Both are GIVEN in the data booklet.
Which points do you use to read a gradient, and how do you get its uncertainty? Two well-separated points on the best-fit line (not the raw data). The uncertainty is Δm = (mₘₐₓ − mₘᵢₙ) ÷ 2 from the steepest and shallowest lines through the error bars.
A graph is a straight line but does NOT pass through the origin — proportional? No. That is linear but not directly proportional. 'Directly proportional' needs the line to pass through the origin (c = 0), i.e. a constant ratio Y/X.
Will repeating and averaging fix a systematic error? No. Averaging only cuts random error (the scatter). A systematic error shifts every reading the same way — you must fix the instrument or method (e.g. correct a zero error).
What are the independent, dependent and control variables? Changed on purpose; measured; kept the same because it could affect the result.
How do you straighten y = A e⁻ᵏˣ? Plot ln y against x: gradient −k, intercept ln A.
What does correcting a zero error do to a graph? It shifts the line; the gradient stays the same, so a gradient-based result is unaffected.
Exam tips
Exam tips
- Match the instrument's RESOLUTION to the quantity, and reading uncertainty = ± half the smallest division. To shrink it, measure a MULTIPLE (a stack of sheets, 10 swings) and divide by the exact count.
- Pick the right propagation rule by the operation: + or − → add ABSOLUTE uncertainties; × or ÷ → add FRACTIONAL/percentage uncertainties; a power aⁿ → multiply the fractional uncertainty by |n|. The last two are in the data booklet.
- In a quotient with a power (like ρ = m ÷ L³), the side's fractional uncertainty counts |n| times — here three times — so that term usually dominates. Measure the powered quantity most carefully.
- Round the absolute uncertainty to 1 significant figure, then match the value to the same decimal place: ρ = 8.0 ± 0.2 g cm⁻³, never 8.00 ± 0.23.
- Read a gradient off TWO well-separated points ON the best-fit line, not the data points. State what the gradient represents and give its units — it is always a physics quantity.
- For the gradient's uncertainty, draw the steepest and shallowest lines that still pass through all the error bars, then Δm = (mₘₐₓ − mₘᵢₙ) ÷ 2.
- Linearize by rearranging the law to Y = mX + c, then plot Y against X. 'Directly proportional' demands a straight line THROUGH THE ORIGIN (or a constant ratio across rows) — a straight line alone is only 'linear'.
- Random error scatters readings → cut it by repeating and averaging. Systematic error shifts them all the same way → averaging won't help; fix the instrument or method. Always discard a clear anomaly before averaging.
- Dimensional analysis: balance the fundamental SI units (kg, m, s, A) on both sides to find an unknown exponent, or to state the units of a constant read off a gradient (y-axis units ÷ x-axis units).
- Curved data: one smooth curve through every error bar; extend it (dashed) only when asked to extrapolate.
- Correct a zero error with its sign: reading − zero reading, so a −0.3 mm zero ADDS 0.3 mm.