🎯 Hook – the meter that never quite agrees with itself
Read a moving-coil ammeter twice and the needle rarely settles on exactly the same mark. Read a digital multimeter twice and the last digit sometimes flickers between two values. Neither meter is broken — every measurement has an uncertainty, a range around the reading within which the true value is expected to lie.
📖 What is uncertainty?
Uncertainty is a quantitative indication of the quality of a result. It is a range of values around a measurement within which the true value is expected to lie, and is an estimate. It is not the same thing as an error: an error arises from the equipment or the technique used and is the actual difference between the reading taken and the true value, whereas the uncertainty is the range you assign around a reading to allow for that possibility. Uncertainties in measurements are recorded as a range (±) to an appropriate level of precision.
| Uncertainty… | …is taken as |
|---|---|
| in a reading | ± half the smallest scale division |
| in a measurement | at least ± 1 smallest division |
| in repeated data | half the range, i.e. ± ½ (largest − smallest value) |
| in a digital reading | ± the last significant digit (unless otherwise quoted) |
🔢 Absolute, fractional and percentage uncertainty
Uncertainty is expressed in one of three forms. Take \(L = 5.0 \pm 0.1\) cm as the running example:
In general:
✍️ Worked example 1 – reading an analogue meter
An analogue milliammeter is read as 1.6 mA. The smallest scale division is 0.2 mA. State the absolute, fractional and percentage uncertainty in this reading.
percentage uncertainty \(= \dfrac{0.1}{1.6}\times100 = 6.2\%\).
⚖️ Propagating uncertainties in processed data
Uncertainty propagates through a calculation differently depending on the operation involved.
| Operation | Example | Propagation rule |
|---|---|---|
| Addition & subtraction | \(y = a \pm b\) | \(\Delta y = \Delta a + \Delta b\) — the sum of the absolute uncertainties |
| Multiplication & division | \(y = ab\) or \(y = a/b\) | \(\dfrac{\Delta y}{y} = \dfrac{\Delta a}{a} + \dfrac{\Delta b}{b}\) — the sum of the fractional uncertainties |
| Power | \(y = a^{n}\) | \(\dfrac{\Delta y}{y} = n\left(\dfrac{\Delta a}{a}\right)\) — \(n\) times the fractional uncertainty |
In short: adding or subtracting measurements → add the absolute uncertainties. Multiplying or dividing measurements → add the percentage (or fractional) uncertainties. Measurements raised to a power → multiply the percentage uncertainty by the power.
📊 Seeing uncertainty in a set of readings
A single number ± its uncertainty is hard to picture — plotting the readings makes the spread visible.
| Display | What's plotted | Shows | How to read it |
|---|---|---|---|
| Dot / scatter plot of trials | Each individual reading, on a number line | How far the readings spread either side of the mean | Half the range (largest − smallest) gives the uncertainty |
| Bar chart with error bars | The mean of each trial, with bar height = ± absolute uncertainty | The size of the reading uncertainty relative to the value | A visually shorter error bar for the same absolute uncertainty means a smaller % uncertainty |
✍️ Worked example 2 – percentage uncertainty in a mean value
A student obtains the following results in an experiment to determine the angular frequency \(\omega\) of a rotating disc. Calculate the percentage uncertainty in the mean value of \(\omega\).
| Reading | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \(\omega\) / rad s⁻¹ | 0.154 | 0.153 | 0.159 | 0.147 | 0.152 |
🔬 Uncertainties and significant figures
Measurements and processed uncertainties must be expressed to an appropriate number of significant figures, because the number of significant figures indicates how precise a measurement is.
In an uncertainty: absolute, fractional and percentage uncertainties should usually be quoted to one significant figure. For example, for \(7.4 \pm 0.2\), the percentage uncertainty is \(\pm2.703\%\), so this is quoted as \(\pm3\%\) (1 s.f.). The main exception is when the first digit is 1 — then the uncertainty can be quoted to two significant figures. For example, for \(36.4652 \pm 5\%\), the absolute uncertainty is \(\pm1.82326\), so this is quoted as \(\pm1.8\) (2 s.f.).
In a quantity: the number of significant figures must match the precision of the absolute uncertainty — the quantity and its uncertainty are quoted to the same number of decimal places, and a final calculated value is rounded so its least significant digit aligns with the uncertainty's place value. For example, \(36.4652 \pm 5\%\) is finally quoted as \(36.5 \pm 1.8\).
In raw data: the number of significant figures should be consistent with the resolution of the measuring instrument.
| Instrument | Resolution | Example reading |
|---|---|---|
| Metre rule | ± 1 mm | 15 ± 1 mm |
| Vernier callipers | ± 0.1 mm | 15.4 ± 0.1 mm |
| Micrometer screw gauge | ± 0.01 mm | 15.42 ± 0.01 mm |
In processed data: calculated values should have no more significant figures than the least precise measurement. For example, for the raw data 5.11, 5.14, 5.12, 5.10, the mean value is 5.1175, which should be quoted to 3 significant figures, i.e. 5.12.
🌍 Real-world: why manufacturers publish a tolerance, not a number
A resistor labelled "10 Ω ± 5%" tells you far more than "10 Ω" alone: it tells you the actual resistance of any resistor from that batch lies between 9.5 Ω and 10.5 Ω. Engineers, and IB examiners, routinely need to decide whether two measured values are genuinely different or whether they simply overlap once their uncertainty ranges are taken into account.
🚀 HL Extension – propagating uncertainty through a multi-step formula
Real formulas often combine several operations at once — for example \(g = 4\pi^2 L / T^2\) for a simple pendulum. Since \(\pi\) is a constant, its uncertainty is zero, so \(4\pi^2\) is treated as an exact number; only \(L\) and \(T\) contribute uncertainty, and \(T\) enters as \(T^2\), so its percentage uncertainty is doubled by the power rule before the division rule combines it with \(L\)'s.
Challenge: if \(L = 1.000 \pm 0.005\) m (0.5%) and \(T = 2.01 \pm 0.02\) s (1.0%), find the percentage uncertainty in \(g\). \(\%\ \text{unc}(g) = \%\ \text{unc}(L) + 2\times\%\ \text{unc}(T) = 0.5\% + 2(1.0\%) = 2.5\%\).
⚠️ Trap corner – common mistakes
| Misconception | Correct view |
|---|---|
| Uncertainty and error mean the same thing | Error is the actual difference between a reading and the true value, caused by equipment or technique; uncertainty is the estimated range assigned around a reading to allow for that |
| A longer decimal answer from a calculator is more accurate | The significant figures in the final answer must match the precision of the absolute uncertainty — extra calculator digits carry no real information |
| Uncertainties should be quoted to as many significant figures as convenient | Quote uncertainties to 1 s.f., unless the leading digit is 1, in which case 2 s.f. is allowed |
| Percentage uncertainties can be added directly for a sum or difference | Percentage/fractional uncertainties are only added for multiplication and division; for addition and subtraction, add the absolute uncertainties |
Complete the statements:
✓ Quoted to 1 s.f.: \(\approx5\%\).
✓ \(\%\ \text{unc}(R) = 3\% + 2\% = 5\%\).
✓ Power rule, \(n=2\): \(\%\ \text{unc}(A) = 2\times1.19\% = 2.38\% \approx 2\%\) (1 s.f.).
✓ \(\pi\) is a constant, so it contributes zero uncertainty.
✓ Result \(= 42 \pm 3\).
✓ A micrometer has a much finer resolution (± 0.01 mm) than a metre rule (± 1 mm), so its readings are quoted to more decimal places.