A length is measured as 0.4382\ m with an uncertainty of 0.005\ m. Rewrite the result to an appropriate number of significant figures. [2 marks]

SEAB Original-style practice: A student measures the diameter of a wire five times with a micrometer, obtaining 0.52, 0.53, 0.52, 0.54 and 0.54\ mm. The micrometer has a known zero error of +0.02\ mm. (a) State whether the zero error is random or systematic, and how it affects accuracy. (b) Determine the best estimate of the true diameter.

(a) A zero error is systematic: it shifts every reading by the same amount in the same direction, reducing accuracy while leaving precision (spread) unchanged. (b) Mean of raw readings: 0.52 + 0.53 + 0.52 + 0.54 + 0.545 = 2.655 = 0.530\ mm. Correct for the zero error by subtracting +0.02\ mm: best estimate = 0.530 - 0.02 = 0.51\ mm. Markers reward identifying the zero error as systematic, the statement that it affects accuracy not precision, the correct mean, and the correction for the zero error in the right direction.

SingaporePhysicsSyllabus dot point

How do random and systematic errors affect a measurement, and how do precision and accuracy describe the quality of data?

Distinguish random and systematic errors, relate them to precision and accuracy, and quote results to an appropriate number of significant figures with an estimated uncertainty

A focused answer to the H2 Physics Measurement learning outcome on errors. Random versus systematic errors, the link to precision and accuracy, and how to quote a measurement with an uncertainty and sensible significant figures.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
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What this dot point is asking

SEAB wants you to tell random and systematic errors apart, to connect them to the ideas of precision and accuracy, and to report a measured value with a sensible uncertainty and an appropriate number of significant figures. These judgements run through Paper 4 (the practical) and the data-based question in Paper 3.

The answer

Random and systematic errors

A random error varies unpredictably from one reading to the next, scattering results above and below the true value. Causes include reaction-time variation, fluctuating conditions and reading the last digit on an analogue scale.

A systematic error shifts every reading by the same amount in the same direction. Causes include a zero error on an instrument, a mis-calibrated scale, or a consistent parallax.

The crucial difference for the exam:

Averaging many repeats reduces random error but cannot remove systematic error.
A systematic error is removed only by finding and correcting its cause (for example, subtracting a zero error).

Precision and accuracy

These two words are not interchangeable.

Accuracy: how close a measurement is to the true value. High accuracy means small systematic error.
Precision: how close repeated measurements are to each other (their spread). High precision means small random error.

A set of readings can be precise but not accurate (tightly clustered around a wrong value because of a zero error), or accurate but not precise (scattered widely but averaging to the true value).

Significant figures and reporting

The number of significant figures in a result should reflect the precision of the measurement. A general rule:

Quote a calculated result to the same number of significant figures as the least precise input.
Quote an uncertainty to one significant figure (occasionally two), and round the value to match its decimal place.

For example $g = (9.78 \pm 0.05)\ \text{m s}^{-2}$ is sensibly reported; $g = (9.78342 \pm 0.05)\ \text{m s}^{-2}$ is not, because the extra digits claim a precision the uncertainty denies.

Estimating the uncertainty of a single reading

For an instrument with smallest scale division $d$ :

A digital instrument: uncertainty is $\pm$ the smallest displayed unit (or as the manufacturer states).
An analogue scale: uncertainty is commonly taken as $\pm \tfrac{1}{2}$ a division, though reaction-dependent measurements (like timing) may warrant more.

For repeated readings, a simple estimate of the uncertainty in the mean is half the range:

\text{uncertainty} \approx \frac{\text{max} - \text{min}}{2}

Worked example

A student times $20$ oscillations of a pendulum three times, recording $33.2\ \text{s}$ , $33.6\ \text{s}$ and $33.1\ \text{s}$ , using a stopwatch reading to $0.1\ \text{s}$ . Find the period of one oscillation with its uncertainty.

Step 1: Find the mean time for 20 oscillations

\bar{t} = \frac{33.2 + 33.6 + 33.1}{3} = \frac{99.9}{3} = 33.3\ \text{s}

Step 2: Estimate the uncertainty in this time

Half the range: $\dfrac{33.6 - 33.1}{2} = \dfrac{0.5}{2} = 0.25\ \text{s}$ , round to $0.3\ \text{s}$ .

Step 3: Find the period of one oscillation

T = \frac{33.3}{20} = 1.665\ \text{s}

Step 4: Scale the uncertainty and report

Dividing by $20$ also divides the absolute uncertainty: $\dfrac{0.3}{20} = 0.015\ \text{s}$ , round to $0.02\ \text{s}$ .

So $T = (1.67 \pm 0.02)\ \text{s}$ . The value is rounded to match the decimal place of the uncertainty.

Examples in context

Example 1. A stopwatch and reaction time. Timing a single short event by hand carries a random error of roughly $\pm 0.2\ \text{s}$ from reaction time. Timing $20$ oscillations and dividing by $20$ shrinks that uncertainty per oscillation by a factor of twenty, which is why practical instructions always ask for multiple oscillations.

Example 2. A mis-set balance. An electronic balance that reads $2\ \text{g}$ with nothing on the pan has a zero error. Every mass it reports is $2\ \text{g}$ too high. The readings may be highly precise (tightly repeatable) yet inaccurate until the balance is tared, which illustrates precision without accuracy.

Try this

Q1. Distinguish between a random error and a systematic error, giving one example of each. [2 marks]

Cue. Random: unpredictable scatter (reaction-time variation). Systematic: constant shift (zero error).

Q2. A length is measured as $0.4382\ \text{m}$ with an uncertainty of $\pm 0.005\ \text{m}$ . Rewrite the result to an appropriate number of significant figures. [2 marks]

Cue. $(0.438 \pm 0.005)\ \text{m}$ , value rounded to match the uncertainty's decimal place.

Q3. A set of readings is described as precise but inaccurate. Explain what this means and suggest one likely cause. [3 marks]

Cue. Readings cluster tightly (small random error) but around the wrong value (large systematic error); likely cause is a zero error or mis-calibration.

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original4 marksA student measures the diameter of a wire five times with a micrometer, obtaining

0.52

0.53

0.52

0.54

and

0.54\ \text{mm}

. The micrometer has a known zero error of

+0.02\ \text{mm}

. (a) State whether the zero error is random or systematic, and how it affects accuracy. (b) Determine the best estimate of the true diameter.

Show worked answer →

(a) A zero error is systematic: it shifts every reading by the same amount in the same direction, reducing accuracy while leaving precision (spread) unchanged.

(b) Mean of raw readings: $\dfrac{0.52 + 0.53 + 0.52 + 0.54 + 0.54}{5} = \dfrac{2.65}{5} = 0.530\ \text{mm}$ .

Correct for the zero error by subtracting $+0.02\ \text{mm}$ : best estimate $= 0.530 - 0.02 = 0.51\ \text{mm}$ .

Markers reward identifying the zero error as systematic, the statement that it affects accuracy not precision, the correct mean, and the correction for the zero error in the right direction.

Original3 marksExplain the difference between precision and accuracy, and describe how taking repeated readings and using a higher-resolution instrument each affect a measurement.

Show worked answer →

Accuracy describes how close a measurement is to the true value; precision describes how close repeated measurements are to one another (their spread).

Taking repeated readings and averaging reduces the effect of random error, improving the reliability of the mean (and revealing the spread), but it cannot remove a systematic error.

Using a higher-resolution instrument reduces the reading uncertainty of each measurement, improving precision, but a higher resolution does not by itself improve accuracy if a systematic error (such as a zero error) is present.

Markers reward correct definitions distinguishing closeness-to-truth (accuracy) from spread (precision), and a clear statement that repeats reduce random error while resolution affects reading uncertainty, with neither removing systematic error.