What relative error measures
Relative error tells you how big a measurement's error is compared with the true value it is trying to hit. An absolute error of 0.5 sounds small if the true value is 1,000, but it is enormous if the true value is 1. Dividing the error by the true value strips out that scale so you can compare accuracy fairly across quantities of any magnitude.
Formally, if the measured (approximate) value is v and the true (exact, accepted) value is v₀, then:
- Absolute error: |v − v₀| — same units as the quantity itself.
- Relative error: |v − v₀| / |v₀| — a unitless ratio.
- Percent error: ( |v − v₀| / |v₀| ) × 100 — the same ratio expressed as a percentage.
The absolute-value bars mean relative error and percent error are always reported as non-negative magnitudes. If you also want the direction of the miss, look at the signed deviation v − v₀: a positive number is an overestimate, a negative number is an underestimate.
A worked example
Suppose the accepted standard-gravity value is 9.81 m/s² and your experiment measures 9.8 m/s². The absolute error is |9.8 − 9.81| = 0.01 m/s². The relative error is 0.01 / 9.81 ≈ 0.0010194, and the percent error is about 0.10194%. So the reading is off by roughly one-tenth of one percent — very accurate.
Why divide by the true value and not the measured value
The standard definition normalizes by the true (accepted) value in the denominator, because that is the fixed benchmark you are comparing against. Some fields, when the true value is unknown, instead divide by the measured value or by the average of the two; those are useful approximations but are not the textbook relative error. If you have an accepted reference, put it in the "true value" field and divide by it.
Common reference points
- 0% error — the measurement equals the true value exactly.
- 1% error — a relative error of 0.01; typical of a decent classroom lab measurement.
- 5% error (0.05) — a common threshold below which many introductory experiments are considered acceptable.
- 100% error (1.0) — the absolute error equals the true value; e.g. measuring 0 when the true value is 50, or 100 when the true value is 50.
Frequently Asked Questions
What is the formula for relative error?
Why can't the true value be zero?
What is the difference between absolute error and relative error?
Why can't the true value be zero?
Is percent error the same as relative error?
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |