Understand measurement uncertainty and how it propagates through calculations
AQA A-Level Physics 1
🎯Distinguish between random and systematic errors
📐Calculate absolute, fractional and percentage uncertainty
➕Combine uncertainties for addition and subtraction
✖️Combine uncertainties for multiplication and division
🔢Raise uncertainties to powers
📊Quote results to appropriate significant figures
Random vs Systematic Errors
All measurements contain errors. Understanding the type of error helps us reduce or account for it.
Random error: Causes measurements to scatter unpredictably above and below the true value. Caused by unpredictable fluctuations in the experiment — e.g. reaction time when using a stopwatch, vibrations, reading a scale. Random errors can be reduced by repeating measurements and averaging.
Systematic error: Causes all measurements to be consistently too high or too low by the same amount. Examples include a zero error on a balance, a mis-calibrated ruler, or heat loss in a calorimetry experiment. Systematic errors cannot be reduced by averaging — you must identify and eliminate the source.
Accuracy: How close a measurement is to the true value. Poor accuracy indicates systematic error.
Precision: How consistent repeated measurements are with each other. Poor precision indicates large random error.
A result can be precise but inaccurate (systematic error shifts all values away from the truth), or accurate but imprecise (wide scatter that happens to average correctly).
Absolute, Fractional and Percentage Uncertainty
For any measurement, we must quote an uncertainty alongside the value. There are three ways to express this:
Absolute uncertainty: Δx (same units as x)
Fractional uncertainty = Δx / x (dimensionless)
Percentage uncertainty = (Δx / x) × 100%
For a single reading with a digital instrument, the absolute uncertainty is usually ±1 in the last digit. For analogue instruments (rulers, ammeters), the uncertainty is typically ±half the smallest scale division.
When you repeat measurements, the absolute uncertainty is taken as half the range: Δx = (xmax − xmin) / 2
Quantity
Value
Abs. uncertainty
% uncertainty
Length
25.4 cm
±0.1 cm
0.39%
Time
3.62 s
±0.01 s
0.28%
Mass
152 g
±1 g
0.66%
Combining Uncertainties: Addition & Subtraction
When quantities are added or subtracted, you add their absolute uncertainties.
If z = x + y or z = x − y:
Δz = Δx + Δy
Note that uncertainties always add — they never cancel. Even for subtraction, the absolute uncertainties sum. This is why measuring a small difference between two large quantities leads to very high percentage uncertainty in the result.
Example: A rod measures 12.4 ± 0.1 cm. A block measures 3.8 ± 0.1 cm. The difference = 8.6 ± 0.2 cm.
When quantities are multiplied or divided, you add their fractional (or percentage) uncertainties.
If z = x × y or z = x / y:
Δz/z = Δx/x + Δy/y
When a quantity is raised to a power n, its fractional uncertainty is multiplied by n (the modulus of n for negative powers):
If z = xⁿ:
Δz/z = |n| × Δx/x
Example: if z = x² and x has 3% uncertainty, then z has 2 × 3% = 6% uncertainty.
When combining multiple operations (e.g. z = x²y / w), add the fractional uncertainties of all terms, applying the power multiplier to each: Δz/z = 2(Δx/x) + Δy/y + Δw/w
Operation
Rule
z = x + y or x − y
Δz = Δx + Δy
z = x × y or x / y
Δz/z = Δx/x + Δy/y
z = xⁿ
Δz/z = |n| × Δx/x
A student measures the diameter of a wire five times: 1.22, 1.20, 1.25, 1.21, 1.23 mm. Find the mean diameter and its absolute uncertainty.
3Absolute uncertainty = range / 2 = 0.05 / 2 = 0.025 mm → round to 1 s.f. → ±0.03 mm
Diameter = 1.22 ± 0.03 mm
A resistor has resistance R = 47 ± 2 Ω. A current I = 0.50 ± 0.02 A flows through it. Calculate the power P = I²R and its percentage uncertainty.
1P = I²R = (0.50)² × 47 = 0.25 × 47 = 11.75 W
2%uncertainty in I = (0.02/0.50) × 100 = 4%
3%uncertainty in I² = 2 × 4% = 8% (power rule)
4%uncertainty in R = (2/47) × 100 = 4.26%
5Total %uncertainty in P = 8% + 4.26% = 12.26% ≈ 12%
6Absolute uncertainty = 12% × 11.75 = 1.41 W ≈ 1 W
P = 12 ± 1 W (2 s.f.)
A micrometer reads 8.40 mm with an uncertainty of ±0.01 mm. A caliper reads 8.30 mm with an uncertainty of ±0.05 mm. Find the difference and its uncertainty.
1Difference = 8.40 − 8.30 = 0.10 mm
2For subtraction: Δ = 0.01 + 0.05 = 0.06 mm
Difference = 0.10 ± 0.06 mm (60% uncertainty — high because the values are close!)
A sphere has radius r = 5.0 ± 0.1 cm. Calculate the volume V = (4/3)πr³ and its absolute uncertainty.
4. A student measures 5 repeat values: 12.1, 12.3, 12.0, 12.4, 12.2 s. What is the absolute uncertainty?
Range = 12.4 − 12.0 = 0.4 s. Absolute uncertainty = range/2 = 0.4/2 = ±0.2 s
5. If z = x³ and x has a fractional uncertainty of 0.04, what is the fractional uncertainty in z?
1. A pendulum's period T is measured by timing 20 complete oscillations, giving 38.4 ± 0.2 s total. The length L = 92.0 ± 0.5 cm. Using g = 4π²L/T², calculate the value of g and its percentage uncertainty.
T = 38.4/20 = 1.92 s; ΔT = 0.2/20 = 0.01 s → %ΔT = 0.52%, %ΔT² = 1.04%. %ΔL = 0.5/92.0 × 100 = 0.54%. g = 4π²×0.920/(1.92)² = 9.83 m s⁻². %Δg = 1.04 + 0.54 = 1.58% ≈ 1.6%. Δg = 1.6% × 9.83 ≈ 0.16 → g = 9.8 ± 0.2 m s⁻²
2. A student claims their measurement of density ρ = 8.9 × 10³ kg m⁻³ is accurate to ±5%. The true density of copper is 8.96 × 10³ kg m⁻³. Comment on whether the result is accurate and whether the uncertainty range is realistic.
5% of 8.9 × 10³ = 445 kg m⁻³. The range is 8.455 × 10³ to 9.345 × 10³ kg m⁻³. Since 8.96 × 10³ lies within this range, the result is consistent with the true value. However, the 5% uncertainty seems large — a careful experiment should achieve 1–2%. The systematic error (if any) is small: (8.96 − 8.9)/8.96 × 100 = 0.67%.
3. Explain why systematic errors cannot be reduced by taking more repeat measurements, whereas random errors can. Suggest one method to identify a systematic error in an electrical resistance experiment.
Random errors scatter both above and below the true value; averaging many readings makes the mean closer to the true value (uncertainty in mean ∝ 1/√n). Systematic errors shift every reading the same way, so the mean is still displaced — averaging cannot correct a bias. To identify systematic error in a resistance experiment: take readings with the voltmeter and ammeter in different circuit configurations (ammeter-external / ammeter-internal) and compare results, or check for a non-zero intercept when plotting V vs I.
4. A student measures v = (2as)^½ where a = 4.0 ± 0.2 m s⁻² and s = 3.5 ± 0.1 m. Find v and its absolute uncertainty.
v = √(2 × 4.0 × 3.5) = √28 = 5.29 m s⁻¹. %Δa = 0.2/4.0 × 100 = 5%. %Δs = 0.1/3.5 × 100 = 2.86%. %Δ(2as) = 5 + 2.86 = 7.86%. Since v = (2as)^(1/2), %Δv = ½ × 7.86% = 3.93%. Δv = 3.93% × 5.29 = 0.208 ≈ 0.2 m s⁻¹. Answer: v = 5.3 ± 0.2 m s⁻¹