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 value | Absolute error | Relative error |
|---|---|---|
| 10 cm | 2 cm | 20% |
| 1 m | 2 cm | 2% |
| 5 m | 2 cm | 0.4% |
| 100 m | 2 cm | 0.02% |
Tips
- State the null and alternative hypotheses before looking at the result.
- Choose the significance level (usually 0.05) in advance.
- Report the effect size and confidence interval, not just the p-value.
- All tests run in your browser using exact distribution functions.
Common mistakes to avoid
- Reading “not significant” as “no effect” – the test may just lack power.
- Running many tests and reporting only the significant ones (use a correction).
- Using a z-test with a small sample and an unknown population standard deviation (use t).
Inputs explained
| Field | What to enter |
|---|---|
| Measured (approximate) value | A number |
| Actual (true or accepted) value | A 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]
Formula
Absolute error = |actual − measured| · relative error = |actual − measured| ÷ |actual| · percent error = (measured − actual) ÷ |actual| × 100%
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.
Frequently asked questions
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.
What formula does the Relative Error Calculator use?
Absolute error = |actual − measured| · relative error = |actual − measured| ÷ |actual| · percent error = (measured − actual) ÷ |actual| × 100%.