relative error
/ REL-uh-tiv EHR-ur /
Missing a target by one metre means nothing until you know how far away it was: one metre off in a free-throw is a disaster, but one metre off when aiming at the Moon is miraculous. Relative error puts an error in perspective by asking not 'how big is the miss?' but 'how big is the miss compared to what I was measuring?' It is the absolute error scaled down by the true value.
Relative error is the absolute error divided by the true (or measured) value, usually expressed as a fraction, a percentage, or in parts per thousand. Because the units cancel in the division, relative error is a pure number with no units, which makes it perfect for comparing the quality of measurements of very different sizes — a milligram and a kilogram can be judged on the same fair scale.
Relative error matters because it is what actually tells you whether a measurement is good enough for its purpose, and it is the natural language of analytical performance (a method 'good to 1%'). The caveat is that relative error blows up when the true value is near zero — dividing a small absolute error by a tiny quantity gives a huge percentage — so for measurements close to zero, the absolute error is the more meaningful and honest figure to quote.
An absolute error of 1 mg is 0.001% when weighing 100 g, but 10% when weighing 0.01 g. The same 1 mg miss is excellent in one case and useless in the other — only the relative error reveals the difference.
The same miss, judged fairly against its scale.
Relative error (measured against the true value) is conceptually distinct from relative standard deviation (measured against the mean of replicates) — the first describes accuracy, the second precision.