Percentage Error and Uncertainty in Measurements
Every physics and chemistry practical asks you to compare a measured value against an accepted or true value, and to report how far off your measurement was — as a percentage. This single skill, done correctly, also underlies how uncertainty combines when you calculate a derived quantity like density from two separately measured quantities. This guide covers both, with four worked examples.
Absolute, relative and percentage error
Relative error = Absolute error ÷ True value
Percentage error = Relative error × 100 = |(Observed − True)/True| × 100
The vertical bars mean "take the positive value" — error is reported as a magnitude, not with a negative sign, unless a question specifically asks whether your measurement was above or below the true value.
Worked example 1 — a straightforward percentage error
The accepted boiling point of water at sea level is 100°C. A student's thermometer reads 98.5°C.
Absolute error = |98.5 − 100| = 1.5°C
Percentage error = (1.5 ÷ 100) × 100 = 1.5%
Worked example 2 — error using an instrument's least count
A student measures the length of a rod as 24.8 cm using a scale whose least count (smallest marked division) is 0.1 cm. The accepted length is 25.0 cm.
Absolute error = |24.8 − 25.0| = 0.2 cm
Percentage error = (0.2 ÷ 25.0) × 100 = 0.8%
Note that the least count itself (0.1 cm) is the instrument's own uncertainty — if no accepted value is given at all, the least count is what you report as the uncertainty in a single reading.
Worked example 3 — how error combines in a product or quotient
Density ρ = mass ÷ volume. Suppose the percentage error in the measured mass is 1% and the percentage error in the measured volume is 2%. What is the percentage error in the calculated density?
For a quantity formed by multiplying or dividing measured quantities, the percentage
errors add (regardless of whether the operation is × or ÷):
Percentage error in ρ = 1% + 2% = 3%
Worked example 4 — error in a quantity raised to a power
The side of a cube is measured as 2.0 cm ± 0.1 cm. Find the percentage error in the calculated volume.
Percentage error in the side = (0.1 ÷ 2.0) × 100 = 5%.
Volume V = (side)³, and for a quantity raised to the power n, the percentage error is
multiplied by n:
Percentage error in V = 3 × 5% = 15%
This rule — multiply the percentage error by the power — is exactly why a small measurement error in a radius produces a much larger error in a calculated volume or area.
Common mistakes that cost marks
- Dropping the modulus (absolute value) sign and reporting a negative percentage error when the question only asks for magnitude.
- Confusing absolute error with percentage error — absolute error carries the same unit as the measurement (cm, °C, g); percentage error has no unit at all.
- Subtracting percentage errors for division — a common but wrong shortcut; both multiplication and division cause percentage errors to add, never subtract.
- Forgetting to multiply by the power when a measured quantity is squared or cubed in the formula (as in Example 4) — this is one of the most frequently tested error-propagation questions.
Where this is tested
| Exam / context | Typical use |
|---|---|
| CBSE/ICSE Class 11–12 Physics practicals | Reporting error in direct and derived measurements |
| JEE Main/Advanced | Error-propagation MCQs — combining errors in products, quotients and powers |
| Chemistry practical file and viva | Justifying the precision of titration and weighing results |
| Lab-based competitive exams | Uncertainty analysis in experimental data questions |
Check your working. Use the calculator suite to verify your division and percentage arithmetic before you write your final answer on an error-analysis question.
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