Measurements and uncertainties

Revision notes

Measurements should be reported with units, a sensible number of significant figures and an uncertainty that reflects the method. Random variation produces scatter and can often be reduced by repeated measurements and averaging. Systematic error shifts measurements consistently and is not removed simply by repetition.

For a measured value x ± Δx, percentage uncertainty is \(\dfrac{100\Delta x}{x}\). When adding or subtracting independent measured quantities, absolute uncertainties are commonly added for a conservative school-laboratory estimate. For products and quotients, percentage uncertainties are added; for a power xⁿ, multiply the percentage uncertainty in x by |n|. Graph gradients and intercepts also carry uncertainty, which can be estimated using acceptable steepest and shallowest lines.

Measurement quality and experimental design

A-level practical work is not just about obtaining a numerical answer; it is about judging how much confidence the evidence deserves. Resolution is the smallest change an instrument can distinguish. Precision describes the spread of repeated measurements or the fineness with which a value is recorded, while accuracy describes closeness to the accepted or true value. A result can be precise but inaccurate if a systematic effect shifts every reading in the same direction.

Random uncertainty produces scatter. Repeats, averaging and measuring a larger interval can reduce its influence. Systematic error produces a consistent bias and is not removed by taking more readings. Typical examples include a zero offset, calibration error or a method that consistently loses heat. When evaluating an experiment, name the actual source of uncertainty rather than using vague phrases such as 'human error'.

Absolute, fractional and percentage uncertainty

An absolute uncertainty has the same unit as the measured quantity. Fractional uncertainty is absolute uncertainty divided by the measured value; percentage uncertainty is the fractional uncertainty multiplied by 100. When a quantity is calculated from multiplication or division, percentage or fractional uncertainties are commonly combined. When a quantity is raised to a power, the fractional uncertainty is multiplied by the magnitude of that power in the usual school-level treatment.

The size of a percentage uncertainty depends on the measured value as well as the instrument. Measuring ten oscillations instead of one, or a long distance instead of a short one, can reduce percentage uncertainty when the absolute reading uncertainty remains similar.

Graphs, gradients and intercepts

A graph can test a model more effectively than a single calculation. Choose variables so the predicted relationship can be recognised clearly. A straight-line graph has the form y = mx + c. The gradient and intercept often correspond to physical constants or combinations of constants, so derive their expected meaning before plotting.

Use a best-fit line rather than joining points. Calculate a gradient from two well-separated points on the best-fit line. If uncertainties are required, maximum and minimum acceptable gradients can be used to estimate gradient uncertainty. An intercept that should theoretically be zero can reveal a systematic offset, but only after considering whether the model itself predicts a non-zero intercept.

Significant figures and reporting

A calculated result should not imply more precision than the data support. Keep additional figures during intermediate calculations, then round at the end. Uncertainty is usually quoted to one significant figure, or sometimes two when the first digit is small, and the measured value should be rounded to the same decimal place.

Evaluating evidence

A strong evaluation links limitation → effect on data → specific improvement. For example, 'heat is lost to the surroundings, reducing the measured temperature rise and causing the calculated specific heat capacity to be too high; improve insulation and reduce transfer time.' This is stronger than simply saying 'use better equipment'.

Exam technique

Separate random scatter from systematic shift. Use units consistently, justify repeats, and explain why a proposed graph should be linear. When asked whether data support a model, refer to the trend, uncertainties, intercept and anomalous points rather than relying on a visual impression alone.

Units, homogeneity and estimates

The seven SI base units are the kilogram, metre, second, ampere, kelvin, mole and candela. Other units are derived from them: the newton is kg m s⁻², the joule kg m² s⁻². Prefixes run from pico (10⁻¹²) and nano (10⁻⁹) to mega (10⁶) and giga (10⁹). An equation is homogeneous if every term has the same base units, which is a useful check on a derivation.

An order-of-magnitude estimate rounds a quantity to the nearest power of ten using sensible assumed values. For differences and sums, absolute uncertainties add. A vernier scale reads to a fraction of the main-scale division by finding the vernier mark that lines up with a main-scale mark.