Error Percentage
Overview
Laboratory practice quantifies measurement accuracy as the absolute deviation from an accepted value, scaled to that value: E% = |measured − actual| ÷ actual × 100. A pendulum experiment returning 9.70 m/s² against the accepted 9.81 m/s² shows 0.11 ÷ 9.81 × 100, about 1.12 percent error. Dividing by the accepted rather than the measured value is deliberate — it keeps the denominator fixed so that repeated trials remain comparable. The absolute-value bars discard direction, which is appropriate for accuracy reporting but hides systematic bias; analysts investigating instrument calibration drop the bars and retain the sign. The quantity is undefined when the accepted value is zero, in which case absolute error in the measured units is reported instead. Percent error measures accuracy against truth, whereas percent difference compares two measurements of unknown correctness.
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $measured$ | Observed value | any | The value produced by the measurement or model. |
| $actual$ | Accepted value | same as observed | The reference or true value; forms the denominator. |
| $E%$ | Percent error | % | Magnitude of deviation relative to the accepted value. |
Quantifying Accuracy Against a Reference
Percent error scales the absolute deviation of a measurement by the value it is compared against:
$$E\% = \frac{|measured - actual|}{actual} \times 100$$
A pendulum experiment returning 9.70 m/s² against the accepted 9.81 m/s² shows $0.11/9.81 \times 100 \approx 1.12\%$ error.
Why the Accepted Value Is the Denominator
Dividing by the accepted rather than the measured value keeps the denominator fixed across repeated trials, so errors from different runs are directly comparable. Using the measured value would make the denominator move with the noise being characterised, which is precisely the quantity under investigation.
Accuracy, Precision and Difference
Three related measures are routinely confused.
| Measure | Compares | Denominator |
|---|---|---|
| Percent error | Measurement against a known truth | Accepted value |
| Percent difference | Two measurements of unknown correctness | Their mean |
| Relative standard deviation | Spread of repeated measurements | Their mean |
Percent error addresses accuracy — closeness to truth. Relative standard deviation addresses precision — repeatability. An instrument can be precise and inaccurate, returning tightly clustered readings around a wrong value.
Signed Error
The absolute-value bars discard direction, which suits accuracy reporting but hides systematic bias. A calibration study drops the bars: a run of consistently positive errors indicates an offset that can be corrected, whereas errors scattered about zero indicate random noise that cannot.
Derivation & History
The measure is the relative error $|\hat{x} - x|/|x|$ of numerical analysis, scaled by 100. Relative error is preferred to absolute error because it is dimensionless and therefore comparable across quantities of different magnitude and unit — a 1 mm error is negligible on a bridge span and fatal on a bearing tolerance. The same normalisation underlies floating-point machine epsilon, defined as the largest relative error introduced by rounding a real number to the nearest representable value.
Worked Examples
Density determination
- Absolute deviation: |2.65 − 2.70| = 0.05
- Divide by the accepted value: 0.05 ÷ 2.70 = 0.01852
- Scale: 0.01852 × 100 = 1.85
Result: 1.85% error
Detecting systematic bias
- Signed errors: +4.0%, +3.0%, +5.0%, +4.0%
- All errors share a sign, so the instrument reads high
Result: ≈ +4% systematic offset, correctable by calibration
Edge Cases & Limitations
Accepted value of zero: The quotient is undefined. Absolute error in the native units is reported instead.
No accepted value exists: For novel measurements percent difference against a second independent method is the available alternative.
Very small accepted values: The denominator amplifies modest absolute deviations into large percentages, which can misrepresent instrument performance near the detection limit.
Comparing across quantities: Percent errors from different physical quantities are dimensionless and therefore comparable, but only if each was computed against a reference of comparable reliability.
Real-World Applications
School and university laboratory reports use it as the standard accuracy statement. Instrument manufacturers quote accuracy as a percentage of reading or of full scale. Forecasting evaluates models with mean absolute percentage error, and metrology laboratories express calibration certificates as relative deviations from a traceable standard.