Relative Error Calculator - Percentage Error Formula
Calculate absolute error, relative error, and percentage error by comparing your observed measurement to the accepted true value.
Enter the true (accepted) value and your observed (measured) value to instantly compute all three error metrics used in science, engineering, and finance.
Relative Error Calculator - Percentage Error Formula
Calculate absolute error, relative error, and percentage error by comparing your observed measurement to the accepted true value.
About the Relative Error Calculator
Relative error, also known as percentage error or fractional error, is a dimensionless measure of how inaccurate a measurement is relative to the size of what is being measured. While absolute error tells you the raw magnitude of the discrepancy between an observed value and a true value, it gives no sense of context. A measurement error of 1 metre is enormous when measuring the height of a room but completely negligible when measuring the distance to the moon. Relative error eliminates this ambiguity by expressing the error as a proportion of the true value, making it universally comparable across measurements of very different magnitudes.
The calculation proceeds in two steps. First, compute the absolute error: take the absolute value of the difference between the observed (measured) value and the true (accepted) value. Second, divide that absolute error by the absolute value of the true value to get the relative error. Multiply by 100 to express it as a percentage. Symbolically: Relative Error = |Observed − True| / |True|, and Percentage Error = (|Observed − True| / |True|) × 100. Note that the true value cannot be zero, since this would make the denominator undefined.
In the physical sciences, relative error is central to assessing experimental accuracy. A physics student measuring the acceleration due to gravity might obtain 9.75 m/s² against the accepted value of 9.81 m/s². The absolute error is 0.06 m/s² and the relative error is 0.06/9.81 ≈ 0.0061, or 0.61%. This small relative error confirms that the experiment was well-designed and carefully executed. If the same student had measured 9.40 m/s², the relative error would be 4.2% — a signal that something in the experimental setup needs investigation.
In manufacturing and quality control, tolerance specifications are almost always expressed as relative errors. A precision machine part with a nominal length of 50 mm and a tolerance of ±0.1 mm has a maximum allowable relative error of 0.2%. Microchip fabrication, aerospace components, and pharmaceutical dosing are examples where extremely small relative errors — often below 0.1% — are required to ensure safety and performance. The relative error calculator makes it straightforward to check whether a measured dimension or dosage is within the specified tolerance band.
In economics and finance, relative error quantifies forecast accuracy. If an analyst predicts a company's quarterly revenue will be $500 million but the actual result is $480 million, the absolute error is $20 million. The relative error is 20/500 = 0.04, or 4%. This 4% figure is far more informative than the dollar figure alone because it allows direct comparison against other forecasts, other companies, and other time periods, regardless of the magnitude of the revenues involved.
One important subtlety is the sign convention. Some formulations of relative error use the signed difference (Observed − True) rather than its absolute value, which preserves information about the direction of the error (over-estimate vs. under-estimate). The calculator here uses the absolute value convention to give a non-negative error magnitude, which is standard in most scientific and engineering contexts. If the sign of the error matters for your application, simply note whether your observed value is larger or smaller than the true value.
Relative error examples
Real-world scenarios across science, engineering, and finance showing how to interpret relative error results.
| Inputs | Results | Context |
|---|---|---|
| True = 10.5 g, Observed = 10.2 g | Abs. Error = 0.3 g, Rel. Error = 0.02857, % Error = 2.857% | Chemistry experiment: a student weighs a compound as 10.2 g against a known mass of 10.5 g. The 2.86% error suggests a small systematic loss. |
| True = 9.81 m/s², Observed = 9.7 m/s² | Abs. Error = 0.11, Rel. Error = 0.01121, % Error = 1.121% | Physics lab: measured gravity is 9.7 m/s² vs. accepted 9.81 m/s². A 1.1% relative error is reasonable for a simple pendulum experiment. |
| True = 50 cm, Observed = 50.1 cm | Abs. Error = 0.1, Rel. Error = 0.002, % Error = 0.2% | Manufacturing: a rod is 0.2% over the nominal length — within tolerance for most general-purpose machining. |
| True = 250000, Observed = 245000 | Abs. Error = 5000, Rel. Error = 0.02, % Error = 2.0% | Financial forecast: quarterly profit was $245k vs. a $250k forecast. The 2% relative error indicates a slightly pessimistic projection. |
How to use the Relative Error Calculator
- Enter the true (accepted) value in the first field. This is the reference value — the theoretically correct, experimentally established, or officially specified quantity.
- Enter the observed (measured) value in the second field. This is the value you actually recorded or calculated in your experiment, measurement, or forecast.
- Click Calculate. The tool instantly displays the absolute error, relative error (as a decimal), and percentage error.
- Check the percentage error against your accepted tolerance. Values below 1% are considered excellent in most science and engineering contexts; values below 5% are typically acceptable.
- Click Reset to clear the fields and start a new calculation, or use the example buttons to load typical real-world values.
Relative Error FAQ
What is the difference between absolute error and relative error?
Absolute error is the magnitude of the raw difference between the observed and true values (|Observed − True|) expressed in the same units as the measurement. Relative error divides that difference by the absolute value of the true value, producing a dimensionless fraction. Relative error is more useful for comparing the accuracy of measurements taken in different units or at different scales.
Can relative error be greater than 1 (or 100%)?
Yes. If the observed value differs from the true value by more than the true value itself, the relative error exceeds 1 (100%). For example, if the true value is 50 and the observed value is 120, the absolute error is 70 and the relative error is 70/50 = 1.4 (140%). This would indicate a very poor measurement or a significantly wrong model.
Why can the true value not be zero?
Relative error is defined as the absolute error divided by the absolute value of the true value. If the true value is zero, the denominator is zero and the division is undefined — mathematically, relative error is infinite. In such cases, absolute error alone must be used to assess the precision of the measurement.
What counts as a 'good' relative error?
The acceptable threshold depends entirely on the application. In precision manufacturing, relative errors must often be below 0.1%. In student physics labs, errors below 5% are typically considered acceptable. In financial forecasting, errors below 2–3% are good. There is no universal standard — always compare against the tolerance or accuracy requirement of your specific context.
Is percentage error the same as relative error?
Percentage error is simply relative error multiplied by 100 to express it as a percentage rather than a decimal. They carry the same information. Relative error = 0.035 is identical to a percentage error of 3.5%. The choice between the two is purely a matter of convention — scientific literature often uses percentage error for clarity.
Does relative error account for systematic vs. random errors?
No — relative error is a summary statistic that measures the total discrepancy between a single observed value and the true value. It does not distinguish between systematic bias (consistent over- or under-estimation) and random noise (fluctuating errors). To separate them, you would need repeated measurements: systematic error can be estimated from the mean of multiple trials, while random error is captured by their standard deviation.