# Relative Error Calculator URL: https://speedycalculator.com/calculators/statistics/relative-error/ Last updated: 2026-10-30 > The relative error calculator compares a measured or approximate value with the true (accepted) value. It gives the absolute error |actual − measured|, the relative error as a fraction and a percentage, the signed percent error that shows over- or underestimation, and the accuracy. ## Inputs - Measured (approximate) value - Actual (true or accepted) value ## Results - Relative error - Absolute error - Relative error as a fraction - Signed percent error (measured − actual) - Direction - Accuracy (100% − relative error) - Relative error in parts per million - Measured ÷ actual ## Formula ``` Absolute error = |actual − measured| · relative error = |actual − measured| ÷ |actual| · percent error = (measured − actual) ÷ |actual| × 100% ``` ## Explanation Being off by 2 cm means something very different for a 5 m beam than for a 10 cm bolt. The absolute error – the size of the difference between a measured or estimated value and the true one – keeps the units of the measurement and does not reveal how serious the mistake is. The relative error divides that difference by the true value, which turns it into a unitless fraction or percentage that can be compared across scales: 2 cm on 5 m is 0.4%, on 10 cm it is 20%. Relative error is used to judge laboratory results against accepted constants, to check the precision of numerical approximations (√2 ≈ 1.41 has a relative error of about 0.3%), to compare forecasts with outcomes and to specify instrument accuracy. The signed version, often called percent error, keeps the direction: a positive value means the measurement was too high, a negative one that it was too low. The true value must not be zero – relative error is undefined there, and close to zero it becomes huge and unhelpful. In that case report the absolute error instead. ### How to use the Relative Error Calculator - **Measured (approximate) value** - **Actual (true or accepted) value** Results update as you type, and everything is calculated in your browser. ### How it works Subtract the actual value from the measured one. The absolute value of the difference is the absolute error; divide it by the absolute value of the actual value for the relative error, and multiply by 100 for a percentage. The signed difference divided by |actual| gives the percent error with its direction. ### Worked example A pendulum experiment gives g = 9.62 m/s², while the accepted value is 9.81 m/s². The absolute error is |9.81 − 9.62| = 0.19 m/s² and the **relative error is 0.19 ÷ 9.81 = 0.01937, or 1.94%**. The signed percent error is −1.94%: the experiment underestimated g, and its accuracy is 98.06%. ### Same absolute error, different relative error True valueAbsolute errorRelative error10 cm2 cm20%1 m2 cm2%5 m2 cm0.4%100 m2 cm0.02% ### Inputs explained FieldWhat to enterMeasured (approximate) valueA numberActual (true or accepted) valueA number ### What each result means - **Relative error** = abs([Actual (true or accepted) value] - [Measured (approximate) value])÷abs([Actual (true or accepted) value])×100 - **Absolute error** = abs([Actual (true or accepted) value] - [Measured (approximate) value]) - **Relative error as a fraction** = abs([Actual (true or accepted) value] - [Measured (approximate) value])÷abs([Actual (true or accepted) value]) - **Signed percent error (measured − actual)** = ([Measured (approximate) value] - [Actual (true or accepted) value])÷abs([Actual (true or accepted) value])×100 - **Accuracy (100% − relative error)** = 100 - abs([Actual (true or accepted) value] - [Measured (approximate) value])÷abs([Actual (true or accepted) value])×100 - **Relative error in parts per million** = abs([Actual (true or accepted) value] - [Measured (approximate) value])÷abs([Actual (true or accepted) value])×1e6 - **Measured ÷ actual** = [Measured (approximate) value]÷[Actual (true or accepted) value] ## Worked example Inputs — Measured (approximate) value: 9.62; Actual (true or accepted) value: 9.81. Results — **Relative error = 1.9368 %**; Absolute error = 0.19; Relative error as a fraction = 0.019368; Signed percent error (measured − actual) = -1.9368 %; Direction = Underestimate – the measurement is too low; Accuracy (100% − relative error) = 98.0632 %; Relative error in parts per million = 19,368 ppm; Measured ÷ actual = 0.980632. ## Key terms - **Absolute error**: |actual − measured|, in the units of the measurement. - **Relative error**: Absolute error divided by |actual|. - **Percent error**: Relative error × 100%, often signed. - **Accepted value**: The reference or true value a measurement is compared with. ## FAQ **How do you calculate relative error?** Divide the absolute error by the true value: |9.81 − 9.62| ÷ 9.81 = 0.0194, or 1.94%. **What is the difference between absolute and relative error?** Absolute error is the difference in the units of the measurement; relative error is that difference as a fraction of the true value, with no units. **Is relative error the same as percent error?** Percent error is the relative error expressed as a percentage, sometimes with a sign showing whether the measurement was too high or too low. **Can relative error be negative?** The relative error itself is an absolute value and never negative; the signed percent error is negative when the measurement is below the true value. **What is the relative error of approximating √2 by 1.41?** |1.414214 − 1.41| ÷ 1.414214 = 0.00298, about 0.30%. **What if the true value is zero?** Relative error is undefined. Report the absolute error instead. **What is a good relative error?** It depends on the task: below 1% is good for school experiments, precision instruments achieve parts per million. ## Sources - [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/) - [NIST – Essentials of expressing measurement uncertainty](https://physics.nist.gov/cuu/Uncertainty/index.html) - [Wolfram MathWorld – Probability and statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html)