Foundations · Measurement
Percent Error Calculator
How far off was a measurement — and is that a lot? Percent error scales the gap between a measured value and the true one against the true value, so a miss of 2 means something very different at a scale of 10 than at 10,000. This gives the absolute, relative and percent error together, keeps the sign so you can see whether you over- or under-shot, and is careful to separate percent error (you have a correct reference) from percent difference (two readings and no referee).
Percent error = |measured − true| ∕ |true| × 100%. Use this when one value is the known, accepted or theoretical answer.
Percent difference = |A − B| ∕ [(A + B) ∕ 2] × 100%. Use this when the two values are on an equal footing and neither is the “correct” one.
Result
In plain English
An error of “2” is meaningless until you know 2 out of what. Percent error answers that by dividing the size of the miss by the true value: being 2 off a true 10 is a 20% error; being 2 off a true 10,000 is a rounding speck. That is the whole job of a relative measure — to make errors at wildly different scales comparable, and to strip away the units so a length error and a mass error can be judged on the same footing.
- absolute error
- The raw size of the miss, |measured − true|, in the original units. Honest but not comparable across scales: 1 cm is a lot on a ruler and nothing on a road.
- relative error
- The absolute error divided by the true value, |measured − true| ∕ |true|. A pure number (no units), so it compares across quantities and scales. 0.02 means “off by one part in fifty.”
- percent error
- The relative error written as a percentage — relative error × 100%. The standard way to report the accuracy of a measurement against a known value.
- percent difference
- For two values where neither is the reference: |A − B| divided by their average, times 100%. Symmetric in A and B — swapping them gives the same answer, unlike percent error.
- sign / direction
- Percent error is usually quoted as a positive magnitude, but the sign of (measured − true) tells you the direction: positive means the measurement read high, negative means it read low.
- accuracy vs precision
- Percent error measures accuracy — closeness to the truth. It says nothing about precision — how repeatable the measurement is. A tight cluster of readings can be precisely wrong.
Frequently asked
How do you calculate percent error?
Percent error = |measured − true| ∕ |true| × 100%. Take the difference between your measured value and the true (accepted) value, drop the sign, divide by the true value, and multiply by 100. For example, a measurement of 10.2 against a true 10 gives |10.2 − 10| ∕ 10 × 100% = 2%. The absolute bars mean the result is normally reported as a positive number; if you want the direction, keep the sign of (measured − true) — here it is positive, so the reading is 2% high.
What is the difference between percent error and percent difference?
Percent error compares a measurement against a known, correct reference and divides by that reference. Percent difference compares two values where neither is privileged and divides by their average, ((A + B) ∕ 2). Use percent error in a lab where the accepted value is known; use percent difference when comparing two independent readings or methods on an equal footing. Percent difference is symmetric — swap the two and you get the same number — whereas percent error changes if you mistakenly treat the wrong value as “true.”
Can percent error be more than 100%, or negative?
Yes to both, depending on convention. It exceeds 100% whenever the absolute error is larger than the true value — a measurement of 5 against a true 2 is a 150% error. It is negative only if you keep the sign instead of taking the absolute value; a signed −4% means the measurement came in 4% below the true value. Most textbooks report the unsigned magnitude, but the signed version carries more information because it shows the direction of the bias.
Why use relative or percent error instead of the absolute error?
Because the absolute error has no context. An error of 5 grams is enormous when weighing a 10 g sample (50%) and negligible when weighing a 10 kg sack (0.05%). Dividing by the true value rescales the error to its context and removes the units, so errors in different quantities and at different magnitudes become comparable. The trade-off: relative error becomes unstable, and then meaningless, as the true value approaches zero — at a true value of exactly zero it is undefined, and you must fall back on the absolute error.