Limitation of Physical Measurements

This guide covers sub-topic 3.1.2 Limitation of physical measurements, the second of three sub-topics in Topic 3.1 Measurements and their errors, from the AQA A-level Physics (7408) specification (first teaching September 2015).

Before studying this

Read Use of SI Units and Their Prefixes first. This sub-topic also underpins the required practical activities threaded through the rest of the course.

Syllabus coverage

AQA A-LEVEL PHYSICS (7408) — Sub-topic 3.1.2

Random and systematic errors. Precision, repeatability, reproducibility, resolution and accuracy. Uncertainty: absolute, fractional and percentage uncertainties represent uncertainty in the final answer for a quantity. Combination of absolute and percentage uncertainties. Represent uncertainty in a data point on a graph using error bars. Determine the uncertainties in the gradient and intercept of a straight-line graph. Individual points on the graph may or may not have associated error bars.

Students should be able to identify random and systematic errors and suggest ways to reduce or remove them. Students should understand the link between the number of significant figures in the value of a quantity and its associated uncertainty. Students should be able to combine uncertainties in cases where the measurements that give rise to the uncertainties are added, subtracted, multiplied, divided, or raised to powers. Combinations involving trigonometric or logarithmic functions will not be required.

Random and systematic errors

A random error causes readings to scatter unpredictably around the true value, and its effect can be reduced by repeating a measurement and averaging. A systematic error shifts every reading consistently in the same direction (for example, a zero-error on an instrument), and averaging repeated readings does not reduce it — it must be identified and corrected, often by recalibrating the instrument or accounting for the offset.

Precision, repeatability, reproducibility, resolution and accuracy

These terms describe distinct aspects of measurement quality. Precision describes how close repeated measurements are to each other. Repeatability is the degree of agreement when the same person repeats a measurement using the same method and equipment. Reproducibility is the degree of agreement when the measurement is repeated by a different person, method, or piece of equipment. Resolution is the smallest change in the quantity being measured that causes a detectable change in the instrument’s reading. Accuracy describes how close a measurement is to the true value.

Uncertainty

Uncertainty expresses the range within which the true value of a measured quantity is expected to lie. It can be expressed as:

• Absolute uncertainty: the uncertainty expressed in the same units as the measurement, e.g. (2.5 ± 0.1) cm
• Fractional uncertainty: the absolute uncertainty divided by the measured value, e.g. 0.1/2.5
• Percentage uncertainty: the fractional uncertainty expressed as a percentage, e.g. 4%

There is a direct link between the number of significant figures given for a quantity and its associated uncertainty — a value quoted to more significant figures implies a smaller relative uncertainty.

Combining uncertainties

When a result is calculated from several measured quantities, the uncertainties must be combined:

• When quantities are added or subtracted, add the absolute uncertainties.
• When quantities are multiplied or divided, add the percentage (or fractional) uncertainties.
• When a quantity is raised to a power n, multiply its percentage uncertainty by n.

Combinations involving trigonometric or logarithmic functions are not required at this level.

Uncertainty on graphs

Uncertainty in a data point can be represented on a graph using error bars, showing the range within which the true value of that point is expected to lie. Error bars may or may not be included for every point on a graph. From a graph, the uncertainty in the gradient and intercept of a best-fit straight line can be determined by comparing the best-fit line with the steepest and shallowest lines that still fit within the error bars (the “worst-fit lines”): draw the steepest and the shallowest lines that still pass through all the error bars, and the uncertainty in the gradient is half the difference between these two gradients. The uncertainty in the intercept is found the same way: read off the y-intercept of the steepest worst-fit line and the y-intercept of the shallowest worst-fit line, and take half the difference between these two intercepts.

Error bars also support a direct comparison between two data points: if their error bars overlap, the difference between the two values may not be significant — the true values could plausibly be equal — and no definite difference should be claimed from the graph alone.

Worked example. A student measures the diameter of a wire five times using a micrometer and obtains: 0.52 mm, 0.54 mm, 0.51 mm, 0.53 mm, 0.53 mm. Calculate the mean value and estimate the absolute uncertainty as half the range.

mean = (0.52 + 0.54 + 0.51 + 0.53 + 0.53) ÷ 5 = 0.526 mm
range = 0.54 - 0.51 = 0.03 mm
uncertainty ≈ half the range = 0.015 mm
diameter = (0.53 ± 0.02) mm

Common mistakes

Confusing random error (reduced by repeating and averaging) with systematic error (not reduced by averaging — must be corrected). Adding percentage uncertainties when quantities are added/subtracted (should add absolute uncertainties instead), or vice versa. Forgetting to multiply the percentage uncertainty by the power when a quantity is raised to a power, e.g. in v² or r³.

Quick revision checklist

• Distinguish random error from systematic error and how each is reduced.
• Define precision, repeatability, reproducibility, resolution and accuracy.
• Express uncertainty as absolute, fractional and percentage.
• Combine uncertainties correctly for addition/subtraction, multiplication/division, and powers.
• Use error bars to find the uncertainty in a gradient and intercept.

Related resources

• Use of SI Units and Their Prefixes — the previous sub-topic
• Estimation of Physical Quantities — the next sub-topic
• AQA A Level Physics hub

This guide is intended to support, not replace, engagement with the official AQA specification and your own teacher’s guidance. Always check the current version of the specification for authoritative detail.