Absolute, Relative and Percentage Error
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.
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.
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.
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.
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.
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.
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.