No measurement is ever perfect. Every reading you take has a little wiggle room, and physics insists you state how much. That range is the uncertainty — and knowing how to record it, convert between its forms, and combine it through a calculation is one of the most heavily-tested practical skills in the whole course.
📚 What you need to know
An uncertainty is the range (±) around a measurement where the true value likely lies
It is not the same as an error (a flaw in equipment or technique)
Three forms: absolute, fractional, and percentage
Adding/subtracting → add the absolute uncertainties
Multiplying/dividing → add the fractional (or %) uncertainties
Raising to a power n → multiply the fractional uncertainty by n
Uncertainties are usually quoted to 1 significant figure
What uncertainty means
An uncertainty is a quantitative estimate of how much a measurement could differ from the true value. If you measure a length as 5.0 ± 0.1 cm, you’re saying the true length is very likely somewhere between 4.9 and 5.1 cm. The ±0.1 cm is the uncertainty.
Crucially, an uncertainty is not an error. An error is a mistake or a flaw — a badly-zeroed instrument, a wrong technique — that pushes your reading away from the truth. An uncertainty is the honest, unavoidable range around any reading, even a careful one.
Where does the ± come from? It depends on how you measured:
Situation
Uncertainty
A single reading (analogue scale)
± half the smallest division
A measurement (two readings, e.g. a length)
at least ±1 smallest division
Repeated data
± ½ (largest − smallest value)
A digital reading
± the last significant digit
Three ways to write it
The same uncertainty can be expressed three ways, and you need to switch between them fluently.
A shorter uncertainty bar means a more precise measurement; a longer bar means a less precise one.
WE 1
A rod is measured as L = 48.0 ± 0.5 cm. Find the percentage uncertainty.
Apply the formula
% unc = (0.5 / 48.0) × 100 = 1.04%Round to 1 significant figure
≈ 1%1%Uncertainties are quoted to 1 sig fig (unless the first digit is 1, when 2 figures are allowed). Here 1% is fine.
Uncertainty in repeated readings
When you repeat a measurement several times, the best estimate is the mean, and the uncertainty is half the range (half the spread between the biggest and smallest reading).
WE 2
Five timings give: 4.62, 4.55, 4.60, 4.58, 4.65 s. Find the mean and its absolute uncertainty.
Step 1 — mean
(4.62+4.55+4.60+4.58+4.65)/5 = 4.60 sStep 2 — half the range
½ × (4.65 − 4.55) = ½ × 0.10 = 0.05 s4.60 ± 0.05 sThe value and its uncertainty are quoted to the same number of decimal places — here two, matching the ±0.05.
Combining uncertainties
When measurements go through a calculation, their uncertainties propagate. Three rules cover almost everything you’ll meet:
Operation
Rule
y = a ± b
Add the absolute uncertainties
y = a × b or a/b
Add the fractional uncertainties
y = an
Multiply the fractional uncertainty by n
WE 3
A rectangle has length 12.0 ± 0.1 cm and width 8.0 ± 0.1 cm. Find its area and the absolute uncertainty.
Step 1 — the areaA = 12.0 × 8.0 = 96 cm²Step 2 — add fractional uncertainties (multiply)
0.1/12.0 + 0.1/8.0 = 0.0083 + 0.0125 = 0.021Step 3 — back to absolute
ΔA = 96 × 0.021 = 2 cm²A = 96 ± 2 cm²For a product you add the FRACTIONS, then multiply by the answer to turn it back into an absolute ±.
WE 4
Kinetic energy is E = ½mv², with m = 2.0 ± 0.1 kg and v = 5.0 ± 0.2 m s−1. Find the percentage uncertainty in E.
Step 1 — mass term (fractional)
0.1 / 2.0 = 0.050Step 2 — speed term (power rule, ×2)
2 × (0.2 / 5.0) = 0.080Step 3 — add them
0.050 + 0.080 = 0.13 = 13%13%Because v is squared, its fractional uncertainty is DOUBLED before adding. The ½ is a constant, so it contributes nothing.
💡 Top tips
Uncertainties are usually quoted to 1 significant figure.
A value and its uncertainty share the same decimal places.
The uncertainty in a constant (like π) is taken as zero.
⚠ Common mistakes
Adding absolute uncertainties when multiplying (add fractional ones)
Confusing an error (a flaw) with an uncertainty (a range)
Forgetting to double the fractional uncertainty for a squared quantity
Quoting an uncertainty to too many significant figures
Mismatched decimal places between a value and its uncertainty
Quick recap: An uncertainty is the ± range around a measurement (not an error). Write it as absolute, fractional, or percentage. Combine by adding absolutes (add/subtract), adding fractions (multiply/divide), or ×n for powers. Quote to 1 significant figure and match decimal places to the value.
Once you can handle uncertainties in numbers, the next step is showing them visually — and turning raw data into a graph you can actually read a result from. Plotting properly, drawing lines of best fit, and pulling gradients and areas off a graph is the skill set coming up in Graphing Skills.
Uncertainties feeling fiddly?
Book a free meeting and we’ll drill the three forms, the combining rules, and the power rule until propagating uncertainty is second nature — exactly what the IA and Paper 1B reward.
