Physics › Measurements › Uncertainty and error

Uncertainty and error

Every measurement comes with an uncertainty, an interval the true value is likely to lie in. This topic covers how to find that interval, how to make it smaller, and how to carry it through a calculation to the final answer.

Year 12AQA 3.1.2 CIE 1.3.1, 1.3.2, 1.3.3 OCR A 1.1.3(c), 1.1.4(d), 1.1.4(e), 2.2.1(a), 2.2.1(b), 2.2.1(c), 2.2.1(d)EDEXCEL 4

Watch Uncertainty and error on the InkPhysics YouTube channel

Builds on SI units and prefixes.

IN THIS TOPIC

Tell a random error from a systematic one, and pick the treatment that actually reduces each.
Use precision, accuracy, resolution, repeatability and reproducibility with the meanings mark schemes use.
Estimate an uncertainty from an instrument's resolution, or from the range of a set of repeats.
Move between absolute, fractional and percentage uncertainty, and combine them through sums, products and powers.
Put error bars on a graph and get the uncertainty in the gradient from the steepest and shallowest lines.

COMMON MISCONCEPTION

A measurement is a number.

Two kinds of error, treated differently

Random errors scatter readings either side of the true value. Electrical noise does it. So does a reaction time that varies from one press of the stopwatch to the next, and so does a draught nudging a balance. Because the direction of the scatter is unpredictable, repeating and averaging genuinely helps, and more repeats pull the mean in tighter, though some random scatter survives any number of them.

FIG. 1The same true value measured two ways. Random error scatters readings around it; systematic error shifts every reading to the same side.

Systematic errors shift every reading the same way. A ruler whose scale starts at 1 mm does it, and so does an ammeter showing 0.02 A with nothing connected. Averaging cannot touch them. Repeat a hundred times and the readings stay consistently wrong, so the only remedy is to find the cause and remove it. Most often the cause is a zero error, and the fix there is the model for all the rest. Check the instrument reads zero when it should, then subtract the offset from every reading you took.

The vocabulary mark schemes use

Precision is how tightly repeated readings cluster, and it says nothing whatever about being right. Accuracy is how close a result sits to the true value. The two are independent, exactly as the two kinds of error predict. A systematic error leaves you precise and wrong. Heavy random scatter leaves you imprecise even when the mean lands on the truth.

FIG. 2Four sets of repeated readings. Tight and centred, tight but displaced, scattered around the centre, and scattered off it: each combination of accuracy and precision is possible.

Three more words carry marks. Resolution is the smallest change an instrument can display, one millimetre for an ordinary ruler. Repeatability means the same experimenter with the same equipment gets consistent results; reproducibility means a different experimenter, or different equipment, still agrees, which is the stronger claim.

Putting a number on the doubt

The uncertainty is the interval within which the true value can reasonably be expected to lie, written after a ± sign. For a single reading, the usual board convention is half the instrument's resolution, so a millimetre ruler contributes ±0.5 mm at each end. Measuring a length means judging both ends, which gives ±1 mm overall. For a set of repeated readings, use half the range, largest minus smallest and then halved.

One uncertainty, three forms. The absolute uncertainty keeps the unit, ±0.5 mm. Divide by the value and you have the fractional uncertainty; multiply that by 100 and you have the percentage.

\text{percentage uncertainty} = \frac{\text{uncertainty}}{\text{value}} \times 100

NOT ON THE AQA DATA SHEET: LEARN IT

WORKED EXAMPLE

From repeats to a quoted result

Four timings of the same fall: 2.31 s, 2.35 s, 2.29 s, 2.33 s. Quote the result with its uncertainty.

The mean is (2.31 + 2.35 + 2.29 + 2.33)/4 = 2.32 s.

Half the range gives the uncertainty, (2.35 − 2.29)/2 = 0.03 s, so quote 2.32 ± 0.03 s.

As a percentage that is 0.03/2.32 × 100 = 1.3%. Notice the value carries no more decimal places than its uncertainty allows.

Combining uncertainties

Calculations mix measured quantities, and three rules combine their uncertainties. Add or subtract quantities, and you add the absolute uncertainties. Multiply or divide, and you add the percentage uncertainties. Raise a quantity to the power n, and its percentage uncertainty is multiplied by n, so a radius known to 2% yields a volume known only to 6%, because volume goes as $r^{3}$ .

The subtraction rule has a consequence worth meeting once. Take $x = 50 \pm 1$ mm and $y = 20 \pm 1$ mm. Their difference is $30 \pm 2$ mm. The value shrank while the doubt grew, so subtracting two nearly equal quantities can leave a percentage uncertainty so large the result is worthless, and well-designed experiments go out of their way to avoid it.

FIG. 3Uncertainty on a graph. Each point carries an error bar; the steepest and shallowest acceptable lines, the extremes still consistent with the bars, bracket the possible gradients.

On a graph, each point's uncertainty becomes an error bar. Draw the best-fit line, then the steepest and the shallowest acceptable lines, the extremes that stay consistent with the error bars. Half the difference between those two gradients is the uncertainty in the gradient, and where the same two lines cut the axis brackets the intercept. Last, let the uncertainty set your significant figures. Quoting 12.4783 when the doubt is ±0.5 claims a precision the experiment never had.

GUIDED PRACTICE

A density budget

A block's mass carries 0.8% uncertainty and its volume 1.5%. Decide which combining rule applies to ρ = m/V, then find the percentage uncertainty in the density.

Show the working

Density divides one quantity by another, so the percentage uncertainties add. 0.8 + 1.5 = 2.3%.

Adding the absolute uncertainties would be meaningless, since kilograms and cubic metres cannot be added at all. The percentage rule exists so that quantities with different units can share one budget.

INDEPENDENT PRACTICE

A power inside a formula

g is measured via g = 2h/t², with 0.5% uncertainty in h and 1.2% in t. Find the percentage uncertainty in g, and the absolute uncertainty if g comes out as 9.7 m s⁻².

Show the working

The t is squared, so its percentage counts twice. 0.5 + 2 × 1.2 = 2.9%.

On 9.7 m s⁻² that comes to 0.029 × 9.7 = 0.28, so quote 9.7 ± 0.3 m s⁻². The uncertainty rounds to one significant figure, and it sets how precisely the value itself may be written.

ASSESSMENT FOCUS

“Repeat and average” earns the mark only for random error. Where the question describes a zero error or a mis-set instrument, averaging is the wrong tool, and the fix to give is finding and removing the offset.
Percentage-uncertainty questions lean on the power rule. Spot the square or the cube before anything else, because a cubed quantity carries three times the percentage uncertainty of the length you measured.
Asked which measurement most needs improving, pick the largest percentage uncertainty. The largest absolute uncertainty is the distractor; the mark is for the largest percentage.
The error-bar procedure is mechanical marks. Steepest and shallowest acceptable lines through the bars, then half the gradient difference.
Match significant figures to the uncertainty. If the doubt sits in the first decimal place, the answer stops there too.
Nothing in this topic appears in the data booklet. Percentage uncertainty, the three combination rules and half-the-range are all recall, so budget revision time accordingly.

CHECK YOURSELF

A wire's diameter is measured as 0.40 ± 0.01 mm. What is the percentage uncertainty in its cross-sectional area?

Show a hint

Area depends on the diameter squared. What does the power rule do to a percentage uncertainty?

Show the answer

Percentage uncertainty in the diameter is (0.01 / 0.40) × 100 = 2.5%.

The area goes as $d^{2}$ , so the power rule doubles that. The area is uncertain by 5%.

Working through the radius changes nothing at all. Halving a value halves its absolute uncertainty too, so the percentage survives the halving untouched.

Averaging reduces random error, though it cannot remove it.

It does nothing to systematic error.

A power multiplies the percentage uncertainty by that power.

WORKBOOK

Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.

19 questions on this topicAnswer them one at a time and mark yourself against the mark scheme.Practise this topic

Or read them with their mark schemes on the uncertainty and error questions page.

17 flashcards on this topicDefinitions, off-sheet equations and a spot-the-error card, scheduled by spaced repetition in your browser.Revise with flashcards

WHERE TO GO NEXT

Uncertainty arithmetic is the maths this lesson leans on, worked through from GCSE.

CHECK YOUR PROGRESS

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Tell a random error from a systematic one, and pick the treatment that actually reduces each.
Use precision, accuracy, resolution, repeatability and reproducibility with the meanings mark schemes use.
Estimate an uncertainty from an instrument's resolution, or from the range of a set of repeats.
Move between absolute, fractional and percentage uncertainty, and combine them through sums, products and powers.
Put error bars on a graph and get the uncertainty in the gradient from the steepest and shallowest lines.

Open the full revision checklist to track your progress across the whole unit.