The Formula
Percent error expresses how far a measurement falls from the accepted value, relative to that value:
Percent error = |experimental − theoretical| / |theoretical| × 100
Dividing by the theoretical value rather than the experimental one is deliberate: the accepted value is the reference, and using the measurement as its own denominator would make the error depend on the mistake.
Percent Error Chart
| Percent error | Generally considered |
|---|---|
| Under 1% | Excellent for most laboratory work |
| 1–5% | Good; typical of careful student experiments |
| 5–10% | Acceptable; worth identifying the error source |
| Over 10% | Suggests a systematic problem with method or equipment |
These bands are conventions, not rules. Analytical chemistry routinely expects under 0.5%; a physics teaching lab measuring gravity with a stopwatch may reasonably accept 10%.
Absolute, Relative and Percent Error
- Absolute error — the raw difference, in the original units. Measuring 9.72 against 9.81 gives 0.09 m/s².
- Relative error — absolute error divided by the accepted value, a dimensionless fraction: 0.00917.
- Percent error — relative error × 100: 0.917%.
Absolute error alone is uninformative without scale. Being 1 cm out matters enormously when measuring a coin and not at all when measuring a road.
Signed or Absolute
Most conventions take the absolute value, since the size of the error is what matters. Keeping the sign is more useful when investigating a systematic bias — a thermometer reading consistently 2% low points to a calibration problem, while errors scattered in both directions point to random noise.
Sources of Error
Systematic errors shift every measurement in the same direction: a miscalibrated instrument, a consistent parallax angle, unaccounted air resistance. Repeating the experiment does not help; only identifying the cause does.
Random errors scatter measurements around the true value: reaction time, small fluctuations, reading between scale marks. Averaging many trials reduces them, in proportion to the square root of the number of trials.
Frequently Asked Questions
What if the theoretical value is zero?
Percent error is undefined — the division fails. Report absolute error instead, which is the standard practice in that situation.
Is percent error the same as percent difference?
No. Percent error compares a measurement against an accepted value. Percent difference compares two measurements of unknown accuracy, and divides by their average rather than by either one.
Can percent error exceed 100%?
Yes, when the measurement is more than double the accepted value. It indicates a serious methodological problem rather than measurement imprecision.