Learning objectives
Distinguish random from systematic error and explain how each affects the precision and the accuracy of a set of repeated measurements
Quote absolute uncertainties from analogue and digital instruments, convert to percentage uncertainty, and propagate uncertainty through sums, differences, products and quotients
Calculate percentage error, compare it with the estimated percentage uncertainty, and judge whether a systematic effect is present
Process data graphically: choose scales, draw a best-fit line, determine a gradient and intercept with units, and use maximum and minimum gradients to estimate an uncertainty
Key definitions
Random error
An unpredictable variation that scatters repeated readings above and below the true value. It limits the precision of the data and is reduced by taking more repeats and averaging.
Systematic error
A fault in the apparatus or the method that shifts every reading in the same direction by a similar amount. It limits the accuracy of the result and is not reduced by repeating.
Absolute uncertainty
The range within which the true value is expected to lie, quoted in the units of the measurement, for example 25.45 ± 0.10 cm³.
Percentage uncertainty
The absolute uncertainty expressed as a percentage of the measured value: (absolute uncertainty ÷ value) × 100.
Percentage error
A comparison of an experimental result with an accepted literature value: (|experimental − accepted| ÷ accepted) × 100. It measures accuracy, not precision.
Best-fit line
A single straight line drawn so that the plotted points are distributed evenly on either side of it. Gradients are taken from this line, never from two raw data points.
Explanation
Two kinds of error. A random error arises from small uncontrolled variations: judging the meniscus, the last digit flickering on a balance, a slightly different reaction time on a stopwatch. Random errors push readings up as often as down, so repeated trials scatter around the mean and the mean itself becomes more reliable as more repeats are taken. A systematic error is built into the apparatus or the method: a balance that reads 0.02 g high with nothing on the pan, a thermometer that reads 1 °C low throughout, heat lost to the surroundings in every combustion trial. Every reading is displaced in the same direction, so the mean is wrong no matter how many repeats are done. Random error therefore governs precision (how closely the repeats agree) and systematic error governs accuracy (how close the mean is to the true value). A set of results can be precise but inaccurate, and occasionally accurate but imprecise.
Quoting an absolute uncertainty. For an analogue scale, such as a thermometer or a measuring cylinder, quote half the smallest scale division; a thermometer graduated every 1 °C gives ±0.5 °C. For a digital display, quote the smallest division the instrument shows, so a balance reading 4.62 g gives ±0.01 g. A burette is graduated every 0.1 cm³, so each individual reading carries ±0.05 cm³. Where a manufacturer prints a tolerance on the glassware, for example a 25.00 ± 0.04 cm³ pipette, use the printed value instead. Convert to a percentage by dividing by the measured value and multiplying by 100; the same absolute uncertainty is much more serious on a small quantity, which is why weighing 0.20 g on a two-decimal balance (±5%) is poor practice while weighing 2.00 g (±0.5%) is not.
Propagating uncertainty. When quantities are added or subtracted, add the absolute uncertainties. A titre is the difference of two burette readings, so its uncertainty is ±0.05 + ±0.05 = ±0.10 cm³, and a temperature rise found from two thermometer readings each ±0.5 °C carries ±1.0 °C. When quantities are multiplied or divided, add the percentage uncertainties: the percentage uncertainty in a concentration found from a mass and a volume is the sum of the two. If a quantity is raised to a power n, multiply its percentage uncertainty by n. Molar masses, the Avogadro constant and other data-booklet values are treated as exact and contribute nothing. Convert back to an absolute uncertainty at the end by taking that percentage of the final answer, and round the uncertainty to one or at most two significant figures.
Percentage error and significant figures. Percentage error compares the experimental result with an accepted value and so reports accuracy. The key evaluative step is to compare the two percentages: if the percentage error is comfortably smaller than the total percentage uncertainty, the apparatus alone accounts for the difference and no further explanation is needed. If the percentage error is much larger, a systematic effect is operating and the evaluation must name it and say in which direction it shifts the result — for example, heat loss to the surroundings makes an experimental enthalpy of combustion less exothermic than the literature value. A calculated result carries no more significant figures than the least precise measurement it used, so a concentration found from 25.0 cm³ and 2.4 g is quoted to two significant figures. Keep extra digits inside the working and round only once, at the end.
Graphs and gradients. Choose scales that use at least half of each axis, label both axes with a quantity and its unit, and plot the independent variable on the x-axis. Draw a single straight best-fit line through the scatter rather than joining the points, and read the gradient from two widely separated points that lie on the line itself. The gradient carries the units of the y-axis divided by those of the x-axis, for example cm³ K⁻¹, and the intercept carries the units of the y-axis. To estimate the uncertainty in a derived quantity, draw the steepest and the shallowest lines that still pass through all the error bars: the uncertainty in the gradient is approximately (maximum gradient − minimum gradient) ÷ 2. A point far from the line is an anomaly; identify it, exclude it from the best-fit line and from any mean, and say why it may have occurred, rather than quietly deleting it.
Designing and evaluating a method. Every investigation has one independent variable that is deliberately changed, one dependent variable that is measured, and a set of controlled variables held constant so that the relationship observed is genuine. Choose apparatus whose precision matches the size of the quantity: a burette rather than a measuring cylinder for a titre, a data logger rather than a stopwatch for a fast temperature change. A strong evaluation is specific. It names the largest single source of uncertainty (found by comparing the percentage uncertainties of each measurement), classifies each weakness as random or systematic, predicts the direction of its effect, and proposes an improvement that actually addresses it, such as insulating the calorimeter and applying a cooling correction rather than simply "repeating more times".
Formulae
percentage uncertainty = (absolute uncertainty ÷ measured value) × 100
sums and differences: add the absolute uncertainties
products and quotients: add the percentage uncertainties
quantity raised to the power n: multiply its percentage uncertainty by n
percentage error = (|experimental value − accepted value| ÷ accepted value) × 100
gradient = Δy ÷ Δx, in units of (y-axis unit) ÷ (x-axis unit)
uncertainty in gradient ≈ (maximum gradient − minimum gradient) ÷ 2
Worked examples
q: A student records an initial burette reading of 1.20 cm³ and a final reading of 26.65 cm³, each with an uncertainty of ±0.05 cm³. Determine the titre with its absolute and percentage uncertainty.
steps: Titre = 26.65 − 1.20 = 25.45 cm³.
The titre is a difference of two readings, so add the absolute uncertainties: ±0.05 + ±0.05 = ±0.10 cm³.
Percentage uncertainty = (0.10 ÷ 25.45) × 100 = 0.39%.
Report the titre as 25.45 ± 0.10 cm³, keeping the two decimal places the burette scale supports.
A 0.39% uncertainty is small, so if the final result differs from the accepted value by several per cent the cause is systematic, not the burette.
In context
A pharmaceutical quality-control laboratory runs every assay in triplicate to expose random scatter, and analyses a certified reference standard alongside the samples so that any systematic drift in the instrument is detected and corrected.
Historical sea-surface temperature records had to be corrected for a systematic error introduced when ships changed from uninsulated canvas buckets to insulated ones, which altered how much the sample cooled before it was measured.
Common misconceptions
Students often write "human error" in an evaluation. In fact a mark is earned only by naming a specific effect, classifying it as random or systematic, and stating the direction in which it shifts the result.
Students often think that repeating a measurement improves accuracy. In fact repeats reduce the random uncertainty in the mean and so improve precision, while a systematic error survives every repeat unchanged.
Students often add percentage uncertainties when subtracting two readings. In fact absolute uncertainties are added for a sum or a difference, and percentage uncertainties only for a product or a quotient.
Students often copy every digit from the calculator. In fact the answer carries no more significant figures than the least precise measurement used to obtain it.
Study tips
State the propagation rule in words before the numbers: "this is a difference, so the absolute uncertainties add" secures the reasoning mark even if the arithmetic slips.
Compare the percentage error with the total percentage uncertainty and say explicitly which is larger; that comparison is what the command term "evaluate" is asking for.
Give every gradient a unit built from the two axes, and take it from two well-separated points that lie on the best-fit line, never from two raw data points.
Identify the measurement with the largest percentage uncertainty before suggesting improvements, so that the improvement you propose targets the dominant weakness.
Reasoning
Uncertainty handling is assessed inside every quantitative topic and carries the largest single share of the internal assessment marks.
Summary
Random error scatters repeats and limits precision; systematic error displaces them all and limits accuracy.
Analogue instrument: half the smallest division. Digital instrument: the smallest division shown.
Add absolute uncertainties for sums and differences; add percentage uncertainties for products and quotients.
Percentage error larger than the percentage uncertainty is evidence of a systematic effect that must be named.
Quote the answer to the significant figures of the least precise measurement used.
Check your understanding
Question 1: Two burette readings each carry ±0.05 cm³. The absolute uncertainty in the titre is…
1. ±0.025 cm³
2. ±0.10 cm³
3. ±0.05 cm³
Question 2: Averaging ten repeated readings will…
1. reduce the random uncertainty in the mean
2. remove a zero error on the balance
3. correct a thermometer that reads 1 °C low
Question 3: A result is calculated from 25.00 cm³, 0.1000 mol dm⁻³ and 2.4 g. It should be quoted to…
1. three significant figures
2. four significant figures
3. two significant figures