Measurements and Errors

Physics is a quantitative science, and every measurement we make carries some degree of uncertainty. Understanding how to measure correctly, express results in the proper units, and account for errors is fundamental to all experimental work at A-Level and beyond. This topic underpins every practical assessment and is assessed explicitly in Paper 3 (AQA) and the Practical Endorsement.

SI Units

The International System of Units (SI) provides a universal framework for measurement. There are seven base quantities, each with a defined SI unit:

Quantity	SI Unit	Symbol
Mass	kilogram	kg
Length	metre	m
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

All other physical quantities are derived from these base units. For example, velocity has units of m s⁻¹, force has units of kg m s⁻², and energy has units of kg m² s⁻² (the joule).

Prefixes

SI prefixes allow us to express very large or very small quantities conveniently:

Prefix	Symbol	Multiplier
tera	T	10¹²
giga	G	10⁹
mega	M	10⁶
kilo	k	10³
milli	m	10⁻³
micro	μ	10⁻⁶
nano	n	10⁻⁹
pico	p	10⁻¹²
femto	f	10⁻¹⁵

Deriving Units for Unfamiliar Quantities

A common exam question asks you to derive the SI base units for a given quantity. The approach is to start from the equation defining the quantity and substitute base units for each term.

Worked Example — Derive the SI base units of the Young modulus.

Young modulus E = stress / strain = (F/A) / (ΔL/L)

Units of F = kg m s⁻²; units of A = m²; strain is dimensionless (m/m).

So units of E = kg m s⁻² / m² = kg m⁻¹ s⁻² (which is equivalent to Pa).

Worked Example — Derive the SI base units of electrical resistance.

R = V/I. Voltage V = W/Q = energy/charge. Energy has units kg m² s⁻²; charge Q = It has units A s.

So V has units kg m² s⁻² / (A s) = kg m² s⁻³ A⁻¹.

Therefore R = V/I has units kg m² s⁻³ A⁻¹ / A = kg m² s⁻³ A⁻² (which is the ohm, Ω).

Exam Tip: When deriving units, always start from the defining equation. Replace every quantity with its SI base units, then simplify. Never try to guess the answer — show every step.

Estimation and Orders of Magnitude

Physicists frequently need to estimate values to check whether an answer is reasonable. An order of magnitude estimate means giving the answer to the nearest power of ten. You are expected to know sensible values for everyday quantities.

Key Values to Know

Quantity	Approximate Value
Mass of an electron	10⁻³⁰ kg
Diameter of an atom	10⁻¹⁰ m
Diameter of a nucleus	10⁻¹⁵ m
Height of a person	2 m
Mass of a car	10³ kg
Speed of sound in air	340 m s⁻¹
Speed of light	3 × 10⁸ m s⁻¹
Atmospheric pressure	10⁵ Pa
Radius of the Earth	6.4 × 10⁶ m
Mass of the Earth	6 × 10²⁴ kg
Distance to the Sun	1.5 × 10¹¹ m

Worked Example — Estimate the number of atoms in a human body.

Mass of a human ≈ 70 kg. The body is mostly water, so we approximate using the mass of a water molecule (about 18 u, where 1 u ≈ 1.66 × 10⁻²⁷ kg).

Mass of one molecule ≈ 18 × 1.66 × 10⁻²⁷ ≈ 3 × 10⁻²⁶ kg.

Number of molecules ≈ 70 / (3 × 10⁻²⁶) ≈ 2 × 10²⁷.

Each water molecule has 3 atoms, so number of atoms ≈ 6 × 10²⁷, i.e. order of magnitude 10²⁸.

Exam Tip: In estimation questions, marks are awarded for a sensible method and a reasonable final answer. Do not waste time being overly precise — round aggressively and focus on powers of ten.

Accuracy and Precision

These two terms have distinct meanings in physics and are frequently confused.

Key Definition: Accuracy describes how close a measured value is to the true or accepted value. An accurate measurement has a small systematic error.

Key Definition: Precision describes how close repeated measurements are to each other. A precise set of measurements has small random error (i.e. low spread).

Understanding the Difference

Imagine a target with a bullseye representing the true value:

Accurate and precise: all shots clustered tightly around the bullseye.
Precise but not accurate: all shots clustered tightly but away from the bullseye (systematic error).
Accurate but not precise: shots scattered widely but centred on the bullseye.
Neither accurate nor precise: shots scattered widely and not centred on the bullseye.

	Small random error (precise)	Large random error (not precise)
Small systematic error (accurate)	Best case — reliable result	Mean is correct but data is spread
Large systematic error (not accurate)	Consistently wrong	Unreliable in every respect

Exam Tip: If asked how to improve accuracy, suggest calibrating the instrument or eliminating systematic errors. If asked how to improve precision, suggest using a more sensitive instrument or taking more repeat readings.

Resolution

Key Definition: Resolution is the smallest change in the quantity being measured that an instrument can detect. For example, a standard ruler has a resolution of 1 mm, while a micrometer has a resolution of 0.01 mm.

Types of Error

Systematic Errors

Systematic errors cause all readings to be shifted in one direction — either all too high or all too low. They do not average out with repeated measurements. Examples include a zero error on a balance, parallax when reading an analogue scale, or a miscalibrated instrument.

Methods to reduce systematic errors:

Calibrate instruments before use
Check for and correct zero errors
Use fiducial markers to reduce parallax
Compare results with accepted values

Random Errors

Random errors cause readings to scatter about the true value. They arise from the limitations of human observation and the inherent unpredictability of the measurement process.

Methods to reduce random errors:

Take repeat measurements and calculate the mean
Use more sensitive measuring instruments
Use techniques such as timing many oscillations to find the period of one

Uncertainty

Every measurement should be quoted with its uncertainty. Uncertainty tells us the range within which the true value is likely to lie.

Sources of Uncertainty

For a single reading, the uncertainty is typically ± half the smallest division (resolution) of the instrument.

For a set of repeated readings, the uncertainty is commonly taken as:

Uncertainty = (max value − min value) / 2

This is called half the range.

Absolute and Percentage Uncertainty

The absolute uncertainty has the same units as the measurement (e.g. length = 25.4 ± 0.2 cm).

The percentage uncertainty expresses the uncertainty as a fraction of the measured value:

Percentage uncertainty = (absolute uncertainty / measured value) × 100%

Combining Uncertainties

When quantities are added or subtracted, their absolute uncertainties are added:

Δ(A ± B) = ΔA + ΔB

When quantities are multiplied or divided, their percentage uncertainties are added:

%(A × B) = %A + %B %(A / B) = %A + %B

When a quantity is raised to a power, the percentage uncertainty is multiplied by that power:

%(Aⁿ) = n × %A

Worked Example — Uncertainty Propagation Through a Multi-Step Calculation

A student measures the following values for a cylindrical wire:

Length L = 0.750 ± 0.001 m
Diameter d = 0.28 ± 0.01 mm (so radius r = 0.14 ± 0.005 mm = 1.40 × 10⁻⁴ ± 5 × 10⁻⁶ m)
Resistance R = 2.15 ± 0.05 Ω

Calculate the resistivity ρ = RA/L = Rπr²/L and its percentage uncertainty.

Step 1: Calculate percentage uncertainties.

% uncertainty in R = (0.05/2.15) × 100 = 2.3%

% uncertainty in L = (0.001/0.750) × 100 = 0.13%

% uncertainty in r = (5 × 10⁻⁶ / 1.40 × 10⁻⁴) × 100 = 3.6%

Step 2: The formula involves r², so double the percentage uncertainty in r.

% uncertainty in r² = 2 × 3.6 = 7.1%

Step 3: Total percentage uncertainty in ρ = %R + %r² + %L = 2.3 + 7.1 + 0.13 = 9.5%

Step 4: Calculate ρ.

A = πr² = π × (1.40 × 10⁻⁴)² = 6.16 × 10⁻⁸ m²

ρ = RA/L = 2.15 × 6.16 × 10⁻⁸ / 0.750 = 1.77 × 10⁻⁷ Ω m

Step 5: Absolute uncertainty = 9.5% of 1.77 × 10⁻⁷ = ±0.17 × 10⁻⁷ Ω m

Final answer: ρ = (1.8 ± 0.2) × 10⁻⁷ Ω m

Exam Tip: The diameter of a wire is often the dominant source of uncertainty because it is small and squared. Always identify which variable contributes the most to the total uncertainty — examiners frequently ask this.

Percentage Error

Percentage error compares your experimental result to the accepted value:

Percentage error = |experimental value − accepted value| / accepted value × 100%

If the percentage error is less than the total percentage uncertainty, the experiment is consistent with the accepted value.

Plotting Graphs and Using Uncertainties

Error Bars

When plotting data on a graph, error bars represent the uncertainty in each measurement. For a data point at (x, y), the error bar extends vertically from (x, y − Δy) to (x, y + Δy), and horizontally from (x − Δx, y) to (x + Δx, y) if both uncertainties are significant.

Described diagram — Graph with error bars: Imagine a scatter plot with data points marked as small crosses or dots. At each point, a short vertical line extends equally above and below the point, showing the uncertainty in the y-value. If x-uncertainties are significant, short horizontal lines extend equally left and right.

Best-Fit and Worst-Fit Lines

The line of best fit passes through or close to as many data points as possible (accounting for error bars), balancing points above and below the line.

The worst-fit line (sometimes called the worst acceptable line) is the steepest or shallowest line that still passes through all the error bars. The difference between the gradient of the best-fit line and the worst-fit line gives the uncertainty in the gradient:

Uncertainty in gradient = |gradient of best fit − gradient of worst fit|

Similarly, the uncertainty in the y-intercept can be found from the difference in intercepts.

Described diagram — Best and worst fit lines: Picture a set of data points with error bars. The best-fit line passes centrally through the data. The worst-fit line is drawn from the top of the first error bar to the bottom of the last error bar (or vice versa), giving the maximum or minimum plausible gradient.

Exam Tip: When drawing lines of best fit, use a sharp pencil and a transparent ruler. The line should pass through the error bars of all points if possible. Anomalous results should be identified and can be excluded — but always note them.

Significant Figures

The number of significant figures in a result reflects the precision of the measurement. When performing calculations, the final answer should be given to the same number of significant figures as the least precise value used.

Rules for Counting Significant Figures

All non-zero digits are significant.
Zeros between non-zero digits are significant (e.g. 305 has 3 s.f.).
Leading zeros are not significant (e.g. 0.0042 has 2 s.f.).
Trailing zeros after a decimal point are significant (e.g. 2.50 has 3 s.f.).

Exam Tip: In intermediate steps of a calculation, keep at least one extra significant figure to avoid rounding errors accumulating. Only round to the appropriate number of significant figures in the final answer.

Summary

Concept	Key Point
SI base units	7 base quantities; all others derived
Estimation	Give answers to nearest power of 10
Accuracy	Closeness to true value (systematic error)
Precision	Closeness of repeated readings (random error)
Absolute uncertainty	Same units as the measurement
Percentage uncertainty	(Δx / x) × 100%
Adding/subtracting	Add absolute uncertainties
Multiplying/dividing	Add percentage uncertainties
Powers	Multiply percentage uncertainty by the power
Error bars	Show uncertainty on graphs
Best/worst fit lines	Determine uncertainty in gradient and intercept