Uncertainty and Error Analysis

Uncertainty and Error Analysis

Understanding uncertainty is essential for evaluating the reliability and validity of experimental measurements. At A-Level, you are expected to calculate, combine, and propagate uncertainties, and to assess their impact on conclusions.

Types of Error

Systematic errors produce measurements that are consistently too high or too low by the same amount. They affect accuracy (closeness to the true value).

Examples:

• A zero error on a measuring instrument
• A thermometer that reads 1°C too high
• Parallax error if the observer consistently views from the same wrong angle
• Background radiation not subtracted in radioactivity experiments

Random errors cause measurements to scatter unpredictably around the true value. They affect precision (spread of repeated measurements).

Examples:

• Timing by hand (reaction time)
• Fluctuations in readings on a voltmeter
• Slight variations in the way a measurement is taken

Accuracy vs Precision

Term	Meaning
Accurate	Close to the true value (low systematic error)
Precise	Measurements closely grouped together (low random error)
Repeatable	Same person gets similar results with same equipment
Reproducible	Different people/equipment get similar results

Measurements can be precise but inaccurate (e.g., consistently wrong due to a systematic error), or accurate but imprecise (scattered around the true value).

Expressing Uncertainty

Absolute uncertainty: ±Δx (same units as the measurement)

Example: length = 25.3 ± 0.1 cm

Fractional uncertainty: Δx/x (dimensionless)

Example: 0.1/25.3 = 0.00395

Percentage uncertainty: (Δx/x) × 100%

Example: (0.1/25.3) × 100 = 0.40%

Estimating Uncertainty from Instruments

Digital instruments: Uncertainty = ±1 in the last displayed digit

Example: Digital ammeter reads 2.34 A → uncertainty = ±0.01 A

Analogue instruments: Uncertainty = ±half the smallest division

Example: Ruler with 1 mm divisions → uncertainty = ±0.5 mm

For a range (e.g., measuring with a ruler): Two readings are taken (start and end), so uncertainty doubles: ±1.0 mm total.

Estimating Uncertainty from Repeated Measurements

Method: Take at least 3 readings and calculate:

Uncertainty = (max value − min value) / 2

This gives the half-range of the data. Alternatively, calculate the mean and standard deviation if many readings are available.

Combining Uncertainties

Addition or subtraction (e.g., c = a ± b):

Δc = Δa + Δb (add absolute uncertainties)

Multiplication or division (e.g., c = ab or c = a/b):

Δc/c = Δa/a + Δb/b (add fractional/percentage uncertainties)

Powers (e.g., c = aⁿ):

Δc/c = n × Δa/a (multiply fractional uncertainty by the power)

Worked Example — Density Calculation

A cylinder has: mass m = 45.2 ± 0.1 g, diameter d = 12.5 ± 0.1 mm, height h = 35.0 ± 0.5 mm.

Density ρ = m / (π(d/2)²h) = m / (πd²h/4)

ρ = 4 × 45.2 × 10⁻³ / (π × (12.5 × 10⁻³)² × 35.0 × 10⁻³)

ρ = 0.1808 / (π × 1.5625 × 10⁻⁴ × 0.0350) = 0.1808 / (1.718 × 10⁻⁵) = 10,520 kg m⁻³

Percentage uncertainties:

• Mass: (0.1/45.2) × 100 = 0.22%
• Diameter: 2 × (0.1/12.5) × 100 = 1.60% (×2 because d is squared)
• Height: (0.5/35.0) × 100 = 1.43%

Total % uncertainty = 0.22 + 1.60 + 1.43 = 3.25%

Absolute uncertainty: 3.25% × 10,520 = ±342 kg m⁻³

Result: ρ = 10,500 ± 300 kg m⁻³ (rounded sensibly)

Note that the diameter contributes the most uncertainty — to improve the experiment, focus on measuring d more precisely (e.g., use a micrometer instead of vernier callipers).

Uncertainty in Graphs

Error bars: Drawn on data points to show the uncertainty in each measurement. Vertical error bars for the y-variable, horizontal for x (often omitted if small).

Line of best fit: Should pass through or close to all error bars (not necessarily through every point).

Worst line (steepest or shallowest line through the error bars): Used to find the uncertainty in the gradient and y-intercept.

Uncertainty in gradient = |best gradient − worst gradient|

Uncertainty in y-intercept = |best intercept − worst intercept|

Significant Figures

• Measurements: Record to the precision of the instrument
• Calculated results: Give to the same number of significant figures as the least precise input data
• Uncertainties: Usually 1 significant figure (occasionally 2 if the leading digit is 1)
• The result should be rounded to match the uncertainty: 10,523 ± 342 → 10,500 ± 300

Identifying the Dominant Uncertainty

In any calculation, the variable with the largest percentage uncertainty is the dominant source of error. Improving the precision of this measurement has the greatest impact on the overall uncertainty. Always identify it and suggest how to reduce it.

Reducing Uncertainty

Strategy	Example
Use more precise instruments	Micrometer (±0.01 mm) instead of ruler (±0.5 mm)
Measure larger quantities	Time 10 oscillations, not 1
Repeat and average	Reduces random error
Use video/data loggers	Eliminates reaction time
Calibrate instruments	Reduces systematic error
Control variables carefully	Reduces scatter