Percentage error quantifies how far off an experimental or estimated value is from the accepted true value. It's a standard accuracy metric in science, engineering, manufacturing, and any field where measurements need to be validated against known standards. A lower percentage error means higher accuracy.
The percentage error formula
The absolute value in the numerator means the result is always positive — it measures the size of the error, not its direction. Some disciplines use a signed version (without the absolute value) to show whether the measurement overshoots or undershoots the true value.
Worked examples
- Gravity experiment: You measure g = 9.75 m/s². The accepted value is 9.81 m/s². Error = |9.75 − 9.81| ÷ 9.81 × 100 = 0.06 ÷ 9.81 × 100 = 0.61%. Excellent accuracy.
- Chemistry lab: You measure the boiling point of water as 99.2°C. Theoretical is 100°C. Error = |99.2 − 100| ÷ 100 × 100 = 0.80%.
- Construction estimate: A contractor estimated a project at $45,000. Actual cost was $52,000. Error = |45,000 − 52,000| ÷ 52,000 × 100 = 13.46%.
- Weather forecast: Predicted temperature of 72°F vs. actual 68°F. Error = |72 − 68| ÷ 68 × 100 = 5.88%.
What counts as acceptable error?
Acceptable error depends heavily on the context:
- Below 1% — Excellent. Expected in precision instruments, physics constants, and pharmaceutical measurements.
- 1% to 5% — Good. Typical for well-controlled lab experiments and professional surveys.
- 5% to 15% — Fair. Acceptable for rough estimates, field measurements, and some engineering tolerances.
- Above 15% — Poor. Usually indicates a systematic error, calibration issue, or flawed methodology.
Percentage error vs. percentage difference
These formulas look similar but serve different purposes. Percentage error compares a measured value to a known true value — one value is the standard. Percentage difference compares two values symmetrically when neither is the accepted truth. Use percentage error for lab results against textbook values. Use percentage difference when comparing two independent measurements.
Sources of error
- Systematic errors — consistent biases from miscalibrated instruments or flawed procedures. These push every reading in the same direction.
- Random errors — unpredictable variations from environmental factors, human reaction time, or instrument noise. These average out over many trials.
- Rounding errors — loss of precision from rounding intermediate calculations too early. Keep extra decimal places during calculations and round only the final result.