Lesson 17: Uncertainty and Error Propagation

Every measurement is wrong — the professional question is *by how much*. This lesson builds the IB's uncertainty machinery (the "core skills" of the guide): absolute and percentage uncertainties, propagation through calculations, error bars and max/min gradient lines. It is assessed in Paper 2's experimental question and throughout the IA.

Types of uncertainty

Random uncertainties scatter readings around a mean (reaction time, judging a meniscus); they shrink with repetition and averaging. Systematic errors shift all readings one way (a mis-zeroed balance, heat loss to the room); repetition does nothing — only a better method or calibration helps. The absolute uncertainty has the same unit as the measurement; the percentage (fractional) uncertainty is the portable form you propagate. The resolution of the instrument (smallest scale division, or ±1 in the last digital digit) is the usual starting estimate.

Propagation rules (data booklet approach)

• Adding or subtracting quantities: add the absolute uncertainties.
• Multiplying or dividing: add the percentage uncertainties.
• Raising to a power n: multiply the percentage uncertainty by n.

Report final answers as value ± uncertainty, with the uncertainty to 1 (or 2) significant figures and the value rounded to the same decimal place.

Worked problem 1 — density with propagated uncertainty

A metal block has mass 54.2 ± 0.1 g and sides 2.50 ± 0.02 cm, 3.00 ± 0.02 cm and 2.00 ± 0.02 cm. Find its density and uncertainty.

Volume V = 2.50 × 3.00 × 2.00 = 15.0 cm³
% uncertainties in sides: 0.02/2.50 = 0.80% ; 0.02/3.00 = 0.67% ; 0.02/2.00 = 1.00%
% uncertainty in V = 0.80 + 0.67 + 1.00 = 2.47%
ρ = 54.2/15.0 = 3.61 g cm⁻³
% uncertainty in ρ = 2.47% + (0.1/54.2 = 0.18%) = 2.65%
Absolute uncertainty = 0.0265 × 3.61 = 0.10 g cm⁻³
ρ = 3.61 ± 0.10 g cm⁻³

The shortest side dominates the budget — measure it with a micrometer, not more care on the mass.

Worked problem 2 — power and the squared term

A current of 1.50 ± 0.02 A flows through a resistor of 8.0 ± 0.1 Ω. Find P = I²R and its uncertainty.

P = 1.50² × 8.0 = 18.0 W
% uncertainty: I contributes 0.02/1.50 = 1.33%, doubled (squared term) → 2.67%
R contributes 0.1/8.0 = 1.25%
Total = 3.9% → absolute uncertainty 0.70 W
P = 18.0 ± 0.7 W

Practical worked example — random vs systematic in the g experiment

Timing a pendulum: 20 swings take 28.4, 28.7, 28.2, 28.5, 28.3 s — the spread (half-range ±0.25 s) is random (reaction at start/stop, varying in sign). Averaging the five totals (mean 28.42 s) shrinks the uncertainty of the mean to roughly ±0.1 s, and dividing by 20 swings shrinks it again: T = 1.421 ± 0.005 s. But if the stopwatch is started late by the same reflex every time, or the string length is measured to the top of the bob (2 cm short on 51 cm → g low by 4%), that offset is systematic — invisible in the spread, visible only against the accepted value. A complete evaluation separates the two and says which limits the conclusion.

Error bars and gradient uncertainty

On graphs, each point carries error bars whose half-length is the uncertainty in each variable. The line of best fit passes through the error bars; maximum and minimum gradient lines — the steepest and shallowest lines still threading all the bars — give the uncertainty in the gradient. If a theoretical value lies between the gradients implied by the max/min lines, the data are consistent with the theory: that sentence is the standard conclusion format for the IA and Paper 2 alike.

Nature of Science

Uncertainty is not an admission of failure; it is what makes a measurement a scientific claim at all. The 2019 redefinition of the kilogram via Planck's constant was possible only when several national laboratories' uncertainty budgets overlapped — metrology is uncertainty management at the highest level.

Real-world application

Drug dosages, bridge load ratings and climate projections are all published as value ± uncertainty, and policy follows the range, not the central figure. An engineer reading "yield strength 250 ± 5 MPa" designs to 245 MPa; the uncertainty is part of the specification.

Common mistakes (examiner style)

• Adding percentage uncertainties for quantities that are added (use absolute there).
• Forgetting to double the percentage for a squared quantity — or tripling wrongly for cubes.
• Averaging away a systematic error ("I repeated it, so the zero error cancels" — it does not).
• Error bars drawn in one variable only when both have significant uncertainties.
• Rounding the value before propagating (keep guard digits until the final statement).

IB Exam Experience

Paper 1-style questions. 1. A length is 4.00 ± 0.05 m. Its percentage uncertainty is: A. 0.0125% B. 1.25% C. 5% D. 0.8%. Key: B — 0.05/4.00 × 100 = 1.25%. 2. The area of a rectangle 3.0 ± 0.1 cm by 2.0 ± 0.1 cm has percentage uncertainty closest to: A. 3% B. 5% C. 8% D. 13%. Key: C — 3.3% + 5.0% = 8.3%. 3. Repeating a measurement and averaging mainly reduces: A. systematic error B. random uncertainty C. resolution limits D. calibration error. Key: B.

Paper 2-style structured question (7 marks). A student measures the resistance of a wire: V = 2.40 ± 0.05 V, I = 0.80 ± 0.02 A. (a) Calculate R and its absolute uncertainty. [4] (b) The same wire's length is 0.500 ± 0.002 m and its diameter 0.40 ± 0.01 mm. Identify the measurement that most limits a resistivity determination, justifying your answer with percentages. [3] *Marking notes:* (a) R = 2.40/0.80 = 3.00 Ω; % uncertainties: V 2.1%, I 2.5% → total 4.6% → ±0.14 → R = 3.00 ± 0.14 Ω (accept ±0.1 or ±0.14 with correct rounding to the same decimal place). (b) Resistivity ρ = RA/L needs area ∝ d²: diameter contributes 2 × (0.01/0.40) = 5.0%, exceeding V (2.1%), I (2.5%) and L (0.4%) — so the diameter, via the micrometre measurement, is the limiting factor; the mark scheme requires the doubled contribution to be recognised.

IA idea: Compare two methods of measuring the same quantity (for example, the density of an irregular solid by displacement vs by calliper geometry) and evaluate which achieves the smaller total propagated uncertainty, and why.

Max/min lines with real numbers

A student plots s against t² for a falling object and reads gradients: best fit 4.95 m s⁻²-worth of gradient (s = ½gt², so g = 2 × gradient), maximum acceptable gradient (steepest line through all error bars) 5.15, minimum 4.72. Then g = 9.90 m s⁻² with a range from 9.44 to 10.30, quoted as g = 9.9 ± 0.4 m s⁻². The accepted 9.81 lies inside the range: the experiment is *consistent* with theory — the strongest honest claim available. Had the range been 9.2–9.6, the correct conclusion is not "close enough" but "a systematic effect is present; candidates are…" — and listing plausible candidates with directions is what earns the evaluation marks.

Significant figures and decimal places: the house rules

Quote the uncertainty to one significant figure (two if the first digit is 1) and match the value's precision to it: 3.61 ± 0.10, not 3.6122 ± 0.1. In intermediate steps, carry at least two extra digits beyond the final precision — premature rounding is a silent mark-loser in multi-part questions where later parts reuse earlier values (error carried forward only rescues method, not a value rounded into the wrong tolerance band). Percentages are usually quoted to two or three significant figures; probabilities and efficiencies between 0 and 1 as decimals or percentages, but never above 100% for an efficiency — a self-check that costs one second.

The pendulum uncertainty budget, complete

For g from T = 2π√(l/g) with l = 0.800 ± 0.002 m and T = 1.794 ± 0.006 s (mean of repeated 20-swing timings): g = 4π²l/T² = 39.48 × 0.800/3.218 = 9.81 m s⁻². Contributions: length 0.25%; T 0.33%, doubled in T² to 0.67%; total 0.92% → Δg = 0.09. Report g = 9.81 ± 0.09 m s⁻². The budget tells you instantly how to improve: timing contributes nearly three times the length term, so time 50 swings instead of 20 (halving the timing contribution to ~0.17%) or use a photogate — decisions driven by arithmetic, not taste.

Chi-squared thinking (awareness only)

Beyond IB requirements but worth one paragraph of awareness: professionals test goodness-of-fit with statistics like chi-squared, comparing squared deviations from a model against the quoted uncertainties. If points scatter much more than their error bars suggest, either the uncertainties were underestimated or the model is wrong — the same judgement you make visually with max/min lines, formalised. When your IA graph shows a best-fit line missing most error bars, that tension is real evidence about the model or the uncertainty estimates, and a perceptive evaluation says which it suspects and how it would test the suspicion (usually: repeat with a changed range).

Reporting a result: the professional format in one line

Everything in this lesson compresses into a single convention: result = (best estimate ± absolute uncertainty) unit, with the relative uncertainty stated or implied. Examples: g = (9.81 ± 0.09) m s⁻² (0.9%); r = (0.85 ± 0.07) Ω (8%); t½ = (4.4 ± 0.2) min (4.5%). A reader can reconstruct your entire error analysis from that line: the central value is your best physics, the uncertainty is your honesty quantified, and the percentage lets other scientists compare your precision against theirs at a glance. Write every final answer — exam or IA — in this format and you are speaking the discipline's native dialect.