Errors in Measurement: Accuracy & Precision

Every measurement is a promise with a margin: "the period is 2.62 s plus or minus 0.11 s". This topic quantifies that margin — the accuracy and precision of instruments, absolute/relative/percentage error, and the propagation rules that turn errors in m and V into an error in density. It is listed explicitly in both the JEE Main syllabus (least count, accuracy and precision, errors in measurement) and the NEET syllabus, even though the rationalised NCERT trimmed its instrument-construction sections (see the chapter hub's declared exclusions). JEE Main asks percentage-error numericals almost yearly; NEET favours accuracy-vs-precision conceptual items. The counting discipline that decides how many digits a result deserves lives on the significant figures page.

What are Errors in Measurement? — Complete Theory

1. Accuracy vs precision (the distinction NTA loves)

Accuracy is closeness of a measurement to the true value. Precision is closeness of repeated measurements to each other — small scatter. The two are independent: a worn metre scale can give beautifully repeatable (precise) readings that are all 2 mm too long (inaccurate). Systematic errors injure accuracy; random errors injure precision. In exam language: "an instrument with small least count gives precise readings; calibration against a standard makes them accurate."

2. Classification of errors

Systematic errors repeat in the same direction and can be traced and corrected: (i) instrumental — a zero error in a screw gauge, a stretched tape; (ii) imperfection in technique — holding the thermometer bulb touching the vessel wall; (iii) personal (bias) — always reading a pointer slightly from the left, parallax; (iv) external conditions — temperature changing a calibrated resistance. Random errors fluctuate unpredictably in sign and size (unsteady hands, voltage jitter) and are reduced by repeating and averaging — statistics is the cure. Gross errors are blunders (misreading 6 as 9) and are removed by care, not analysis.

3. Least count — the built-in limit

The least count (LC) is the smallest value an instrument can resolve: LC = (value of one main-scale division) ÷ (number of divisions on the coinciding sub-scale). Vernier calliper: LC = 1 mm ÷ 10 = 0.1 mm. Screw gauge of pitch 0.5 mm with 50 circular divisions: LC = 0.5/50 = 0.01 mm = 10 μm. Every single reading is uncertain by ±(about) half the least count, which is exactly why the last digit of a reading is the "uncertain" significant figure of significant-figure theory.

4. Absolute, mean absolute, relative and percentage error

Let a₁, a₂, …, aₙ be n readings of quantity a. Define:

a_mean = (a₁ + … + aₙ)/n — the best estimate (averaging tames random error).
Absolute error of each reading: Δaᵢ = |aᵢ − a_mean| — always positive, always a magnitude.
Mean absolute error: Δa_mean = (Δa₁ + … + Δaₙ)/n. The result is reported as a = a_mean ± Δa_mean.
Relative (fractional) error: Δa_mean/a_mean — dimensionless, which is why it survives into every derived quantity.
Percentage error: (Δa_mean/a_mean) × 100%.

5. Combination (propagation) of errors — the three rules

Visualising Errors in Measurement

Think of each measurement as an arrow shot at the true value. The four-target figure is the whole theory in one picture: tight clusters mean small random error (precision), on-centre clusters mean small systematic error (accuracy). The propagation rules then act like a funnel: every multiplication adds another fractional-error stream, and powers multiply the streams. When you compute ρ = m/V with Δm/m = 2% and ΔV/V = 3%, picture 2% and 3% pouring into one funnel to give 5% — never 1%, never (2% + 3%)/2.

Solved Examples on Errors in Measurement (Step-by-Step)

Solved Example 1 — Mean, absolute and percentage error (NCERT-style data)

Q. The period of a pendulum is measured five times: 2.63 s, 2.56 s, 2.42 s, 2.71 s, 2.80 s. Report T with its error and the percentage error.

Step 1 (mean). Sum = 2.63 + 2.56 + 2.42 + 2.71 + 2.80 = 13.12 s. Mean = 13.12/5 = 2.624 s → round to the readings' precision: 2.62 s.

Step 2 (absolute errors). |2.62 − 2.63| = 0.01; |2.62 − 2.56| = 0.06; |2.62 − 2.42| = 0.20; |2.62 − 2.71| = 0.09; |2.62 − 2.80| = 0.18. Sum = 0.54 s.

Step 3 (mean absolute error). ΔT = 0.54/5 = 0.108 s → 0.11 s (rounding to the error's own precision).

Step 4 (percentage). (0.11/2.62) × 100 ≈ 4.2% → quote as ≈ 4%.

Solved Example 2 — Propagation into a derived quantity

Q. The percentage errors in measuring mass and volume of a body are 2% and 3%. Find the percentage error in the density ρ = m/V.

Step 1. ρ = mV⁻¹ — a product with one power of −1.

Step 2. Δρ/ρ = Δm/m + ΔV/V = 2% + 3% = 5%.

Step 3. Note the "−1" power of V would matter for a quantity like V² (error × 2), but for V itself the fractional error passes through as-is.

Practice Questions on Errors in Measurement (With Solutions)

Practice Q1 (Numerical-value type)

Q. In T = 2π√(l/g), the percentage errors in l and T are 2% and 3%. Find the percentage error in g.

Solution. g = 4π²l/T² → Δg/g = Δl/l + 2·ΔT/T = 2% + 2 × 3% = 8%. The 4π² is exact and contributes nothing.

Practice Q2 (Single-correct MCQ)

Q. If Z = A − B with ΔA = 0.2 cm and ΔB = 0.1 cm, then ΔZ = (a) 0.1 cm (b) 0.3 cm (c) 0.2 cm (d) cannot be found

Solution. Absolute errors add for both sum and difference: ΔZ = 0.2 + 0.1 = 0.3 cm. Subtracting B cannot "cancel" its uncertainty — errors are magnitudes.

Practice Q3 (Assertion–Reason)

Q. Assertion (A): An instrument can give highly precise readings that are systematically inaccurate. Reason (R): Precision concerns the scatter of repeated readings, not their closeness to the true value.

Solution. Both true: a zero-error screw gauge repeats beautifully yet reads high every time. R is exactly the conceptual reason A is possible — R explains A.

Key Formulas & Takeaways

FAQs on Errors in Measurement

What is the difference between accuracy and precision?

Accuracy is closeness to the true value; precision is the agreement among repeated measurements (small random spread). A precise instrument can still be inaccurate if a systematic error is present.

How are errors combined in Z = A + B and Z = AB?

For sum or difference the absolute errors add: ΔZ = ΔA + ΔB. For a product or quotient the fractional errors add: ΔZ/Z = ΔA/A + ΔB/B — and for powers the fractional error is multiplied by the exponent.

What is percentage error?

The mean absolute error expressed as a fraction of the mean value and multiplied by 100 — (Δa_mean/a_mean) × 100%.

What is a systematic error? Give one example.

An error that repeats in the same direction in every reading — e.g. a zero error in a screw gauge, a worn scale, or personal bias in reading a pointer; it is reduced by calibration, not by repeating readings.