No measurement is ever perfect. A ruler can only be read so finely, a stopwatch depends on your reflexes, a meter needle sits somewhere between two marks. Rather than pretend the doubt away, physics states it openly: every result comes with an uncertainty — a ± range within which the true value almost certainly lies. Handling those ranges correctly is what separates a raw number from a trustworthy result, and it’s a skill your Internal Assessment rewards heavily.
📘 What you need to know
Uncertainty is an estimate of the range around a measurement where the true value lies — not the same as an error
Three forms: absolute (the ± amount), fractional (a ratio), and percentage (that ratio × 100)
Reading uncertainty is ± half the smallest division on an analogue scale, or ± the last digit on a digital display
Repeated data: uncertainty is half the range — ½ (largest − smallest)
When you add or subtract, add the absolute uncertainties
When you multiply or divide, add the fractional (or %) uncertainties
For a power, multiply the fractional uncertainty by that power
Quote absolute uncertainties to 1 significant figure, and match the value’s decimal places to it
The three kinds of uncertainty
The same doubt can be expressed three ways, and each is useful at a different moment. The absolute uncertainty is the raw ± amount in the measurement’s own units. The fractional uncertainty divides that by the value, giving a pure ratio. The percentage uncertainty is simply the fractional one as a percentage.
For a length measured as L = 5.0 ± 0.1 cm, the absolute uncertainty is 0.1 cm, the fractional is 0.1 ÷ 5.0 = 0.02, and the percentage is 2%.
Reading uncertainty off an instrument
Where does the ± come from in the first place? It depends on the instrument. On an analogue scale, you can reasonably read to half the smallest division; on a digital readout, the uncertainty is the last displayed digit. A meter needle sitting between the marks is the classic case.
The needle falls between marks, so the reading is ± half the smallest division — here ± 0.1 mA.
Situation
Uncertainty to use
a single reading
± half the smallest division
a measurement (two readings)
at least ± 1 smallest division
repeated data
half the range: ± ½ (largest − smallest)
digital readings
± the last significant digit
Combining uncertainties
Once a result comes from several measurements, their uncertainties have to be carried through the calculation. The rule you use depends on the operation.
Operation
Example
Rule
add or subtract
y = a ± b
add the absolute uncertainties: Δy = Δa + Δb
multiply or divide
y = ab or a÷b
add the fractional uncertainties: Δy÷y = Δa÷a + Δb÷b
raise to a power
y = an
multiply the fractional uncertainty by n: Δy÷y = n(Δa÷a)
+ or −
→ add →
absolute uncertainties
× or ÷
→ add →
fractional uncertainties
A memory hook that never fails me: plus/minus → add the pluses-and-minuses (the absolute amounts), while times/divide → add the percentages. And for powers, the exponent just scales the percentage — cubing something triples its percentage uncertainty. Get those three straight and you can propagate anything the exam throws at you.
Quick recap: uncertainty is a ± range; read it as half a division (analogue) or the last digit (digital); add absolute uncertainties for ±, add fractional/% for ×÷, and multiply the fractional by the power for an.
Quoting uncertainties properly
The final polish is presentation. An absolute uncertainty is quoted to 1 significant figure (with a rare exception when the first digit is 1). The measured value is then rounded so its last decimal place lines up with the uncertainty — it would be absurd to write 2.4173 ± 0.1.
🧭 Presenting a result
Round the uncertainty to 1 significant figure
Round the value so its final digit sits in the same place as the uncertainty
State the units on both the value and the absolute uncertainty
Add the percentage too, if the question asks for it
WE 1
A student times one swing of a pendulum five times and records: 2.41, 2.38, 2.44, 2.40 and 2.42 s. Find the mean period and its absolute and percentage uncertainty.
Mean period
(2.41 + 2.38 + 2.44 + 2.40 + 2.42) ÷ 5
= 12.05 ÷ 5 = 2.41 sUncertainty (half the range)
± ½ (2.44 − 2.38)
= ½ × 0.06 = 0.03 sPercentage
(0.03 ÷ 2.41) × 100 ≈ 1.2%
T = 2.41 ± 0.03 s (≈ 1%)Uncertainty quoted to 1 s.f., and the mean given to the same 2 decimal places.
WE 2
The potential difference across a resistor is 6.0 ± 0.2 V and the current through it is 1.5 ± 0.1 A. Calculate the resistance R = V ÷ I and its absolute uncertainty.
This is a division, so add the fractional uncertainties, then convert back to an absolute uncertainty at the end.
Resistance
R = V ÷ I = 6.0 ÷ 1.5
= 4.0 ΩAdd fractional uncertainties
ΔV÷V = 0.2÷6.0 = 0.033
ΔI÷I = 0.1÷1.5 = 0.067
ΔR÷R = 0.033 + 0.067 = 0.10 (10%)Convert to absolute
ΔR = 0.10 × 4.0 = 0.4 Ω
R = 4.0 ± 0.4 Ω (10%)
💡 Top tips
Match the operation to the rule: ± uses absolute uncertainties, ×÷ uses fractional ones — never mix them
Powers scale the percentage: a squared term doubles its % uncertainty, a cube triples it
Constants like π contribute zero uncertainty — ignore them in the propagation
Round the uncertainty to 1 s.f. first, then round the value to match its place value
⚠ Common mistakes
Adding absolute uncertainties in a multiplication (or fractional ones in a sum) — the operation decides the rule
Forgetting to convert the fractional result back to absolute before quoting ±
Quoting an uncertainty to too many figures — it’s 1 s.f., so 0.03, not 0.0300
Leaving the value more precise than the uncertainty, e.g. 2.413 ± 0.03
Up next: Graphing in Physics — we’ll turn tables of data into graphs, draw lines of best fit, read gradients and intercepts, and see how linearising a curve unlocks the physics hidden inside it.
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