Percent Error Calculator
How far a measurement falls from the accepted value, as a percentage.
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Your details
Result
- Percent error
- 0.611621%
- Signed relative error
- -0.611621%
- Absolute error
- 0.06
- Percent difference
- 0.613497%
Always positive — magnitude only
The measurement is low
Against the mean of the two, when neither is the accepted value
9.75 against an accepted 9.81…
% error = |measured − accepted| ÷ |accepted| × 100
measured − accepted = 9.75 − 9.81 = -0.06
|-0.06| ÷ |9.81| = 0.00611621
% error = 0.611621%
Formula — written step by step
Measured 9.75 against an accepted 9.81…
% error = |measured − accepted| ÷ |accepted| × 100
measured − accepted = 9.75 − 9.81 = −0.06
|−0.06| ÷ 9.81 = 0.006116
0.006116 × 100
% error = 0.611621%
The accepted value goes in the denominator, because the error is expressed relative to the truth rather than to what you happened to measure. Taking the absolute value makes the result a magnitude, with direction reported separately.
A worked example
The same absolute error, two contexts
- Measurement A
- 10.5 mm where the truth is 10.0 mm
- Measurement B
- 1000.5 mm where the truth is 1000.0 mm
- Both are off by exactly 0.5 mm
- A: 0.5 ÷ 10.0 = 5% error
- B: 0.5 ÷ 1000.0 = 0.05% error
- A hundred-fold difference in relative terms
5% against 0.05%
This is why relative error is reported rather than absolute. Half a millimetre out on a component the size of a fingernail is a serious defect; the same half millimetre over a metre is excellent precision. The absolute figure alone cannot distinguish them.
How to use this calculator
- Enter what you measured, then the accepted or theoretical value.
- Read the percent error as a magnitude — it is always positive by convention.
- The signed row tells you the direction: positive means your measurement was high.
- Use percent difference instead when neither value is the accepted one — comparing two measurements of equal standing, for example.
Error relative to the truth
Percent error expresses how far a measurement falls from an accepted value, as a proportion of that accepted value. Dividing by the truth rather than the measurement is deliberate: it keeps the denominator fixed, so repeated measurements of the same quantity are directly comparable.
The result is conventionally reported as a positive magnitude, because in most contexts the size of the discrepancy is what matters. Where the direction matters too — a scale that consistently reads high, for instance — the signed value is the more informative figure, and both are shown here.
Accuracy and precision are different
Percent error measures accuracy: closeness to the true value. It says nothing about precision, which is the repeatability of a measurement. The two are independent, and confusing them is the most common error in reporting experimental results.
An instrument can be precise but inaccurate — returning 9.50, 9.51 and 9.50 when the truth is 9.81 — which points to a systematic bias such as a miscalibration. It can also be accurate on average but imprecise, scattering widely around the right answer. The first is usually fixable; the second needs better technique or better equipment.
When percent error will not work
The formula fails when the accepted value is zero, because dividing by zero has no meaning. This is not a quirk of the arithmetic: relative error genuinely has no interpretation when there is nothing to be relative to. Report the absolute error instead.
It also becomes unstable when the accepted value is merely close to zero, where a tiny absolute discrepancy produces an enormous percentage. Temperatures near 0 °C are the classic trap — and a reminder that on an arbitrary scale like Celsius, percent error is not a meaningful quantity at all.
Formula & methodology
What this calculator assumes
- Divides by the accepted value, so it is undefined when that value is zero and unstable when it is near zero.
- Measures accuracy, not precision. An instrument can be highly repeatable and consistently wrong.
- Meaningless on an interval scale with an arbitrary zero, such as Celsius, where a percentage of a temperature has no physical interpretation.
Sources
Error measures, and when each applies
| Measure | Formula | Use when |
|---|---|---|
| Absolute error | |measured − accepted| | You need the size in original units |
| Percent error | |diff| ÷ |accepted| × 100 | One value is the accepted truth |
| Signed relative error | diff ÷ |accepted| × 100 | Direction of the bias matters |
| Percent difference | |diff| ÷ mean × 100 | Neither value is authoritative |
The choice of denominator is what distinguishes these. Percent error divides by the truth; percent difference divides by the average of the two, because there is no truth to divide by.
Frequently asked questions
Should percent error be positive or negative?
By convention it is reported as a positive magnitude. The signed relative error is shown separately when the direction of the discrepancy matters.
What is the difference between percent error and percent difference?
Percent error compares a measurement against an accepted truth. Percent difference compares two values of equal standing, dividing by their mean because neither is authoritative.
What counts as an acceptable percent error?
It depends entirely on the field. School experiments often accept under 10%; analytical chemistry may require under 1%; some physical constants are known to parts per billion.
Why can I not use this when the accepted value is zero?
Because the formula divides by it. Relative error has no meaning without a non-zero reference — report the absolute error instead.