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.
The edge lands between 2.5 and 3.0 cm, never exactly on a mark, so the reading you record is an estimate. Move the object: wherever it stops the true edge falls between two divisions and you can only approximate.
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.
The true value is 50 and you read 49: 1 below, but the gap is still 1. Absolute error is |measured − true| = 1. Switch the reading: 49 and 51 give the same gap because absolute error counts size only, not direction.
Relative error: putting it in proportion
Absolute error alone can be misleading, which is why relative error exists. An error of 1 centimetre means very different things on a 10 centimetre pencil and along a 10 metre classroom wall. 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, 1 centimetre out of 10 is a relative error of 0.1; along the wall, 1 centimetre out of 1000 is just 0.001. 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 10 m classroom wall.
The same 1 cm error sits on a pencil of 10 cm, so the gold slice is 1 ÷ 10 = 0.1 of the whole. Pick a longer object and watch the relative error shrink: the bigger the true value, the less a fixed error matters.
Percentage error: the intuitive form
Percentage error is simply relative error expressed as a percentage, found by multiplying the relative error by 100. The pencil's relative error of 0.1 becomes 10%, while the wall's 0.001 becomes 0.1%. Percentages are often the most intuitive form: saying a measurement is off by 2% communicates its quality instantly. For a true value of 50 with an absolute error of 1, the relative error is 1/50 = 0.02, and the percentage error is 2%.
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 the relative error times 100: 0.02 × 100 = 2%. The gold slice fills exactly that share of the whole bar. Choose a bigger relative error and both the percentage and the slice grow together.
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 47 millimetres might really be anywhere from 46.5 to 47.5 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, 0.05 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.
On a scale read to the nearest 1 mm, a reading of 47 mm could be anywhere across the shaded band, so the largest the absolute error can be is half that division, 0.5 mm. Choose a finer scale and the band narrows: more precision means a smaller built-in error.
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 far too large when marking out a 100 metre race track, where it would put the finish line 2 metres out, 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 builder marking out a 5 metre wall and a machinist cutting a 20 millimetre pin may both be out by half a millimetre, but that is 0.01 percent of the wall and 2.5 percent of the pin, so it matters enormously to the machinist and not at all to the builder. 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 a race track, so error is always judged in context.
The error is fixed at 2%, yet for cutting timber the verdict is acceptable: a 2% slip on a 30 cm piece is about 6 mm, easily trimmed. Switch the job and the same number flips between fine and unacceptable, because error only means something once you ask what it is for.
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 100 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?