Q.If the measured values of two quantities are A ± ΔA and B ± ΔB, ΔA and ΔB being the mean absolute errors. What is the maximum possible error in A ± B? Show that if $Z = A/B$, then $\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$.

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). …