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Add Percentage Errors for Products and Quotients, Multiplying by the Power

Every measurement has an uncertainty, and a calculated quantity inherits uncertainty from each input. The rules for combining them are short: add absolute errors for sums and differences, add percentage errors (each multiplied by its power) for products, quotients and powers.

NSEJS · Units, Dimensions and Measurement · 14 linked questions · 100% coverage

Know this first

Measurement, Significant Figures and Error Propagation is the central idea in this question pattern. Identify the condition in the stem before applying the method below.

Study deeper: NCERT Science · Units, Dimensions and Measurement

Reading instruments and significant figures

Leading zeros are not significant, while trailing zeros after a decimal point are: 0.007 has 1 significant figure, 2.64 x 10²⁴ has 3, 0.0006032 has 4 and 6.3200 has 5. A stopwatch of least count 1/5 s measuring 25 s has a maximum error of 0.2/25 = 0.8%. For a screw gauge the reading is MSR + (circular reading x LC) minus the zero error, with its sign: 5 mm + 25 x 0.001 cm = 0.525 cm, and a zero error of -0.004 cm means subtracting a negative number, which adds 0.004 cm, giving 0.529 cm.

Mean values and absolute errors

The mean of 1.34, 1.38, 1.32 and 1.36 is 1.35, the deviations are 0.01, 0.03, 0.03 and 0.01 so the mean absolute error is 0.02, and the percentage error is 0.02 / 1.35 = 1.48%. For periods 2.63, 2.56, 2.42, 2.71 and 2.80 s the mean is 2.624 s and the mean of the absolute deviations is about 0.11 s. Many readings reduce random errors, but they cannot remove a systematic error such as defective weights or a zero error.

Propagation rules

For X = M^a L^b T^c, the percentage error is (a α + b β + c γ)%, each percentage multiplied by its power

R = V/I with V = 10 ± 0.5 V and I = 2 ± 0.2 A has errors 5% + 10% = 15%, so R = 5 ± 15% Ω

A free-fall measurement

g = 2h/t² has \Delta g/g = e1 + 2 e2

For g = 4π² ℓ / T², Δg/g = Δℓ/ℓ + 2ΔT/T;

measuring 100 oscillations (about 200 s) with a 0.1 s stopwatch gives ΔT/

T = 0.05%, so with ℓ ± 1 mm out of 100 cm the total is 0.1% + 0.1% = 0.2%

For ε = x y² / (10 z^(1/3)), with errors 2%, 1% and 3%, the result is 2 + 2(1) + (1/3)(3) = 5%

For a cylinder volume

V = π D² ℓ / 4 with D = 4.00 cm (± 0.01) and ℓ = 5.0 cm (± 0.1), ΔV/V = 2(0.01/4.00) + 0.1/5.0 = 2.5%

OlympIQ exam lens

✓Count significant figures: leading zeros no, trailing zeros after a decimal point yes
✓Least-count error = LC / reading; screw gauge = MSR + CSR x LC - zero error (with sign)
✓Sums and differences: add absolute errors; products and quotients: add percentage errors
✓Powers: multiply the percentage error by the exponent
✓Many readings reduce random error but not systematic error

OlympIQ trap

Do not subtract percentage errors for a quotient. Errors add in both products and quotients, since the worst case is always the largest possible total deviation, so R = V/I carries the sum of the two percentage errors.

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