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Error questions are among the most predictable in Class 11 physics, yet students drop marks because they learn three rules without knowing when each applies. This post gives the rules, a full pendulum example and the usual slips.
Is this still on the syllabus?
Yes for entrance exams. The JEE (Main) 2026 syllabus lists "least count, significant figures, errors in measurements", and the NEET (UG) 2026 syllabus uses the same line. JEE Advanced 2026 asks for "methods of measurement and error analysis" for named experiments, including the simple pendulum and vernier and screw gauge readings. The current NCERT Class 11 chapter on units and measurement keeps significant figures but treats error combination only briefly, in a worked example of a sheet's area, so most practice comes from other sources. Check your exam's bulletin for the current year.
What are absolute, relative and percentage errors?
Absolute error: the size of the uncertainty, in the same units as the measurement. A length of 25.3 ± 0.2 cm has an absolute error of 0.2 cm.
Relative error: absolute error divided by the value: 0.2/25.3 ≈ 0.008.
Percentage error: relative error × 100, here about 0.8 per cent.
For repeated readings, take the mean as the value and the mean of the absolute deviations as the absolute error. Five timings of 2.01, 1.98, 2.00, 2.03 and 1.98 s have a mean of 2.000 s. The deviations are 0.01, 0.02, 0.00, 0.03 and 0.02 s, with a mean of 0.016 s, so T = 2.00 ± 0.02 s.
Which errors do you add, and when?
Rule 1: sums and differences add absolute errors. If Z = A + B or Z = A − B, then ΔZ = ΔA + ΔB.
Example: 25.3 ± 0.2 cm minus 12.1 ± 0.1 cm is 13.2 ± 0.3 cm. Notice that the percentage error has grown from under 1 per cent in each reading to about 2.3 per cent in the difference. Subtracting two close numbers is where precision is lost.
Rule 2: products and quotients add relative errors. If Z = AB or Z = A/B, then ΔZ/Z = ΔA/A + ΔB/B.
NCERT's own example: a sheet measures 16.2 ± 0.1 cm by 10.1 ± 0.1 cm. The relative errors are about 0.6 per cent and 1 per cent, so the area 163.62 cm² carries about 1.6 per cent, or ± 2.6 cm². NCERT then quotes the result as 164 ± 3 cm².
Rule 3: powers multiply the relative error. If Z = Aⁿ, then ΔZ/Z = |n| ΔA/A. A cube's volume has three times the percentage error of its side. Negative and fractional powers follow the same rule, using the size of n.
Worked example: g from a simple pendulum
A student measures L = 100.0 ± 0.1 cm and times 20 oscillations in 40.0 ± 0.2 s. Find g and its error, using g = 4π²L/T².
Period: T = 40.0/20 = 2.000 s. The timing error is shared by 20 oscillations, so ΔT = 0.2/20 = 0.010 s.
Relative errors: ΔL/L = 0.1/100.0 = 0.1 per cent. ΔT/T = 0.010/2.000 = 0.5 per cent.
Combine: Δg/g = ΔL/L + 2ΔT/T = 0.1 + 2 × 0.5 = 1.1 per cent. The 4π² is exact and adds nothing.
Value: g = 4π² × 1.000/(2.000)² = 9.870 m/s².
Absolute error: 1.1 per cent of 9.870 ≈ 0.11 m/s², so g = 9.87 ± 0.11 m/s².
The period dominates because it is squared, and timing many oscillations divides the reaction-time error.
A second example with a cube power: density of a sphere, ρ = m/(4πr³/3). With m = 50.0 ± 0.1 g (0.2 per cent) and r = 1.50 ± 0.01 cm (about 0.67 per cent), Δρ/ρ = 0.2 + 3 × 0.67 ≈ 2.2 per cent. The radius error dominates again.
How do professional scientists combine errors?
The rules above give the maximum possible error. The international Guide to the Expression of Uncertainty in Measurement (JCGM 100:2008), published through BIPM, combines independent uncertainties in quadrature instead. For a product of powers like g, its clause 5.1.6 gives the relative combined uncertainty as the square root of the sum of squares of each power times its relative uncertainty. For the pendulum that is √(0.1² + 1.0²) ≈ 1.0 per cent, slightly smaller than 1.1 per cent, because independent errors rarely all push the same way. Use the adding rule in school and entrance exams unless a question says otherwise.
Common mistakes and how to avoid them
Subtracting errors in a difference or a quotient. Errors always add.
Forgetting the power. T² doubles the timing error; r³ triples the radius error.
Adding absolute errors in a product. Convert to relative errors first.
Ignoring the number of oscillations. Divide the timing error by it.
Quoting too many digits. Round the error to one or two significant figures, then round the value to the same decimal place. Our guide to dimensional analysis as a fast check catches a different kind of slip.
Students' intuitions about measurement are less settled than they look. In a study at the University of Cape Town, Allie and colleagues (1998) probed first-year science students' ideas about repeating measurements and the spread of data, and found that their reasoning depended on context, with intuitive ideas about precision and accuracy that were not kept distinct.
If you understand the rules but freeze on a long formula, the Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can see where each error term comes from. For instrument readings themselves, see our guide to reading a vernier calliper and screw gauge.
A practice routine for this topic
Day 1: write the three rules and one example of each from memory.
Day 2: five sum, difference and product questions.
Day 3: five questions with powers, including one with a square root and one with a negative power.
Day 4: two full experiment questions: the pendulum and a density measurement.
Day 5: redo Day 4 with quadrature and compare the two answers.
Today, redo the pendulum example timing 50 oscillations instead of 20 and see how much smaller the error in g becomes.
