Hong Life CalculatorsUnderstanding the percent error calculator — concept and principle
Percent error measures how far a measured or experimental value is from an accepted true or theoretical value, expressed as a percentage of the true value. It is the standard way to judge the accuracy of a lab result: a student timing a pendulum to estimate gravitational acceleration, for example, compares the result with the accepted value of about 9.81 m/s².
How it is calculated
- Absolute error: |measured − true|, in the same unit as the inputs.
- Percent error: |measured − true| ÷ |true| × 100.
- Signed relative error: (measured − true) ÷ true × 100. A negative sign means the measurement was too low, a positive sign too high.
Both values must be in the same unit. The true value cannot be zero, because the error is expressed relative to it.
The principle
An error of 0.21 means little on its own: it is large for a quantity near 1 and tiny for a quantity near 1,000. Dividing by the true value turns the raw error into a relative one, so results of very different sizes can be compared on the same footing. The absolute value in the usual formula reports only the size of the miss; the signed version keeps the direction, which helps reveal a systematic bias such as a stopwatch that always starts late.
Worked example
- Inputs: measured value 9.6, true value 9.81 (both in m/s²).
- Absolute error = |9.6 − 9.81| = 0.21.
- Percent error = 0.21 ÷ 9.81 × 100 = 2.1407…% ≈ 2.14%.
- Signed relative error = (9.6 − 9.81) ÷ 9.81 × 100 ≈ −2.14%, so the measurement was about 2.14% too low.
| True / theoretical | Percent error |
|---|---|
| 2.45 | 291.44% |
| 4.91 | 95.72% |
| 7.36 | 30.48% |
| 9.81 | 2.14% |
| 12.3 | 21.71% |
| 14.7 | 34.76% |
| 19.6 | 51.07% |
How to read the result
Smaller is better: 0% means perfect agreement. What counts as acceptable depends on the method and instruments, so compare the percent error with the expected uncertainty of the experiment rather than with a universal cut-off. If the percent error is much larger than the instrument precision suggests, look for a systematic cause such as calibration, reaction time or a wrong formula.
Common mistakes
- Dividing by the measured value instead of the true value.
- Forgetting the absolute value and reporting a negative "percent error" where a size is expected.
- Using a rounded reference value with more precision than it has; standard gravity, for instance, is defined as 9.80665 m/s², and 9.81 is only a rounding.
Frequently asked questions
Can percent error be greater than 100%?
Yes. If the measurement is more than twice the true value, or of the wrong sign, the error exceeds 100%.
What is the difference between percent error and percent difference?
Percent error compares a measurement with a known reference. Percent difference compares two values when neither is the reference, dividing by their average.
Is percent error the same as accuracy?
It is one way to express accuracy, meaning closeness to the true value. Precision is different: it describes how close repeated measurements are to each other.
Sources: JCGM 100:2008, Guide to the expression of uncertainty in measurement (GUM) · JCGM 200:2012, International vocabulary of metrology (VIM), entry "measurement error"