Relative Error Calculator

Enter a measured (approximate) value and the true (exact) value to get the relative error, percent error, and absolute error, with the formula shown step by step.

Quick Facts

Formula
Relative error = |measured − true| / |true|
Multiply by 100 for percent error. The result is unitless.

Your Results

Calculated
Relative error
-
|measured − true| / |true|
Percent error
-
Relative error × 100
Absolute error
-
|measured − true|, in original units
Signed deviation
-
measured − true (sign shows direction)

Ready

Enter a measured value and a true value, then press Calculate.

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?
Relative error = |measured − true| / |true|. The measured value is your reading or approximation; the true value is the accepted or exact value. Multiply the result by 100 to get percent error. The numerator on its own, |measured − true|, is the absolute error.
What is the difference between absolute error and relative error?
Absolute error is |measured − true| and keeps the original units (e.g. 0.01 m/s²). Relative error divides that by the true value, giving a unitless ratio (0.00102) or a percentage (0.102%). Relative error lets you compare accuracy across measurements of very different sizes, which absolute error cannot.
Why can't the true value be zero?
Relative error divides by the true value, and division by zero is undefined. When the true value is zero, relative and percent error have no meaning — report the absolute error |measured − true| instead, or normalize by a characteristic scale of the problem.
Is percent error the same as relative error?
Yes — percent error is simply relative error multiplied by 100 and written with a "%" sign. A relative error of 0.03 is a percent error of 3%. Both are always reported as a magnitude (non-negative) because of the absolute-value bars in the formula.

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