ACARA v9 CONTENT DESCRIPTION “calculate and interpret absolute, relative and percentage errors in measurements, recognising that all measurements are estimates”
Builds on: the Year 9 Measurement strand. This unit builds on percentages, ratio, and decimal place value. Understanding measurement error underpins honest reporting in science and the rates, ratio and scale modelling that completes the measurement strand.
Every measurement is an estimate
Every measurement is an estimate. A ruler, a kitchen scale, or a stopwatch can only ever get close to a true value, never pin it down perfectly, because every instrument has a limit to how finely it can read. Accepting that measurements are approximations, rather than exact truths, is the starting point of this unit, and the three kinds of error it introduces are simply different ways of saying how close an estimate is.
Measurement is an estimate
A ruler can only be read to its nearest mark, so an edge that falls between two divisions forces you to estimate; every measurement is an approximation, never the exact truth.
A ruler can only be read to its nearest mark, so a true edge usually falls between two divisions and the reading you record is an estimate. Every measurement works this way: a close approximation, never the exact truth.
Absolute error: the size of the gap
The first is absolute error, the most direct measure of how far off a reading is. It is the size of the gap between the measured value and the true value, taken as a positive amount regardless of direction. If a true length is fifty centimetres and you measure forty-nine, the absolute error is one centimetre; measuring fifty-one would also give an absolute error of one centimetre, since only the size of the discrepancy matters, not whether you went over or under. Absolute error carries the same units as the measurement itself, so here it is one centimetre.
Absolute error: the gap
Absolute error is the size of the gap between a reading and the true value, taken as positive: a reading of 49 or 51 against a true 50 is one unit out either way.
On a number line the true value is 50. A reading of 49 and a reading of 51 are each one unit away, so the absolute error |measured − true| is 1 in both cases. Absolute error measures only the size of the gap and ignores direction.
Relative error: putting it in proportion
Absolute error alone can be misleading, which is why relative error exists. An error of one centimetre means very different things on a ten centimetre pencil and on a ten metre running track. Relative error puts the absolute error in proportion by dividing it by the true value, producing a pure number with no units. On the pencil, one centimetre out of ten is a relative error of nought point one; on the track, one centimetre out of a thousand is just nought point zero zero one. The same absolute error is a hundred times more significant in the first case, and relative error is what reveals this.
Same error, different proportion
Dividing the same 1 cm error by the true value shows how significant it is: a tenth of a short pencil but only a thousandth of a long track.
The same 1 cm error fills a tenth of a 10 cm pencil but only a thousandth of a 1000 cm track. Dividing the error by the true value gives a relative error of 0.1 for the pencil and 0.001 for the track, a hundred times more significant on the pencil.
Percentage error: the intuitive form
Percentage error is simply relative error expressed as a percentage, found by multiplying the relative error by one hundred. The pencil's relative error of nought point one becomes ten percent, while the track's nought point zero zero one becomes nought point one percent. Percentages are often the most intuitive form: saying a measurement is off by two percent communicates its quality instantly. For a true value of fifty with an absolute error of one, the relative error is one over fifty, which is nought point zero two, and the percentage error is two percent.
From relative to percentage
Multiplying a relative error by 100 turns it into a percentage: 0.02 becomes 2 percent, the form most people find easiest to judge.
Percentage error is just the relative error written as a percentage: 0.02 × 100 = 2%. The bar shows 2% of the whole, a small but easily understood share, which is why percentages are the most intuitive way to report error.
Where error comes from: instrument precision
Where do these errors come from in the first place? A large part is the precision of the instrument. A ruler marked only in millimetres cannot distinguish anything finer, so a length recorded as forty-seven millimetres might really be anywhere from forty-six and a half to forty-seven and a half millimetres. The largest the absolute error could be is therefore half a millimetre, half of the smallest division on the scale. A digital scale reading to the nearest tenth of a gram carries a maximum error of half of that, nought point zero five grams. Recognising this built-in uncertainty is part of reading any instrument honestly.
Instrument precision
The precision of an instrument sets a built-in limit: a ruler marked in millimetres can be out by up to half a division, so the maximum absolute error is 0.5 mm.
A ruler marked in millimetres can place a reading of 47 mm anywhere from 46.5 to 47.5 mm, so the true value sits inside the shaded band. The largest the absolute error can be is 0.5 mm, half the smallest division on the scale.
Interpreting error in context
Interpreting an error matters as much as calculating it. A two percent error might be perfectly acceptable when cutting a length of timber, yet completely unacceptable when dispensing medicine, so the same number can be good or bad depending on context. Comparing two measurements is often a question of relative or percentage error rather than absolute error: a baker measuring flour and an engineer machining a part may both be out by a gram, but the gram matters enormously to the engineer and not at all to the baker. The numbers only become meaningful once you ask what the measurement is for.
Context decides acceptability
The same percentage error can be good or bad depending on the job: 2 percent is fine for timber but not for medicine, so error is always judged in context.
A 2% error is perfectly acceptable when cutting timber but not when dispensing medicine. The same number earns a different verdict depending on what the measurement is for, which is why error must always be interpreted in context.
A routine for the three errors
A clear routine ties the three together. First find the absolute error as the positive difference between the measured and true values, keeping its units. Then divide by the true value to get the relative error, a unitless proportion. Finally multiply by one hundred for the percentage error, the form most people find easiest to judge. Throughout, remember the founding idea: because every measurement is an estimate, quoting a result without any sense of its error tells only half the story. Knowing how large the error is, and how significant that size is in context, is what turns a raw reading into a trustworthy measurement.
Quick self-check
1. A true length is 50 cm and you measure 49 cm. What is the absolute error?
2. For a true value of 50 with an absolute error of 1, what is the relative error?
3. An absolute error of 3 on a true value of 60 is what percentage error?
4. An error of 2 cm occurs on an 8 cm object and on a 200 cm object. Where is it more significant?
5. A ruler is marked to the nearest millimetre. What is the maximum absolute error in a single reading?