Every measurement carries an error — not a mistake, but the unavoidable gap between the measured value and the true value, set by the instrument and the method. This concept is about quantifying that gap for a single quantity and then tracking how it grows when several measured quantities are combined into a result. At JEE Main the second part — error propagation — is where almost all the marks are.
1 — Accuracy vs precision. Accuracy is closeness to the true value; precision is closeness of repeated readings to each other (a small spread), regardless of whether they cluster around the truth. A finer least count improves precision; removing a zero error improves accuracy. The two are independent — readings can be precise but inaccurate (a consistent systematic offset) or accurate on average but imprecise.
2 — Types of error. Systematic errors are one-sided and reproducible (instrumental/zero error, imperfect technique, personal bias, environmental drift) — they are corrected, not averaged away. Random errors scatter unpredictably either way and are reduced by averaging many readings. Gross errors are outright mistakes (a misread or mis-recorded value).
3 — Absolute, relative and percentage error. From n readings, the mean is the best estimate. The absolute error of a reading is |reading − mean|; the mean absolute error Δa is the average of these. The relative (fractional) error is Δa / a_mean, and the percentage error is that × 100. If a quantity is measured just once, its absolute error is taken as the instrument's least count.
4 — Propagation of errors (the heart of it). When measured quantities combine, their errors combine by fixed rules, always taking the worst case (magnitudes add — never cancel):
- Sum or difference
Z = A ± B: the absolute errors add, ΔZ = ΔA + ΔB. (Even for a difference — and the percentage error of a small difference can blow up.)
- Product or quotient
Z = AB or A/B: the relative errors add, ΔZ/Z = ΔA/A + ΔB/B.
- Powers
Z = Aᵖ Bᵍ / Cʳ: each exponent multiplies its relative error, ΔZ/Z = |p|·ΔA/A + |q|·ΔB/B + |r|·ΔC/C (the denominator term is still added).
5 — Applied composite formulae. These rules are examined through real experiments: density ρ = m/V (so %ρ = %m + 3·%L for a cube of side L), g = 4π²L/T² from a pendulum (%g = %L + 2·%T, with T timed over n oscillations to shrink the timing error), R = V/I, Young's modulus Y = FL/(πr²ΔL) (%Y = %F + %L + 2·%r + %ΔL), and so on. The dominant contribution is the term with the largest exponent × its own %error — not necessarily the raw largest %error, because the exponent re-weights it.
A dimensional shortcut. The exponents in the propagation formula are exactly the powers in the quantity's dimensional formula: force [MLT⁻²] → %F = %M + %L + 2·%T; energy [ML²T⁻²] → %M + 2·%L + 2·%T.
How this concept is examined. JEE Main asks for the mean/percentage error of a data set, the maximum permissible error of a computed result, which measured quantity dominates the error, or how averaging/least count reduces it. The skill is mechanical but unforgiving: pick the right rule (absolute for sums, relative for products), weight each term by its exponent, and add magnitudes for the worst case.