Calculate Relative Error: Formula and Examples

Calculate relative error by dividing the absolute difference between a measured value and the accepted reference value by the absolute value of that reference. Enter both values below to get relative error as a decimal, percentage, and absolute difference.

Formula

Relative error = absolute error ÷ absolute reference value. As a percentage, multiply that ratio by 100:

Relative error (%) = |measured value − reference value| ÷ |reference value| × 100

• Measured value (m): the result you observed or calculated. It has the same unit as the reference value.
• Reference value (r): the accepted, known, or comparison value, in the same unit as the measurement.
• Absolute error: |m − r|, the nonnegative difference between the two values. Its unit is the same as the values being compared.
• Relative error: the absolute error divided by the absolute reference value. It is a ratio without units; multiply by 100 to express it as a percentage.

The absolute-value signs make the error magnitude nonnegative, whether the measured value is above or below the reference. Some contexts report signed relative error instead, using (m − r) / r × 100 to show direction. Check which convention is required before comparing results.

Worked Example

Using the calculator’s defaults, the measured value is 9.8 and the reference value is 10. Both are in the same unspecified unit.

1. Find the absolute difference: |9.8 − 10| = 0.2.
2. Divide by the absolute reference: 0.2 ÷ |10| = 0.02.
3. Convert the ratio to a percentage: 0.02 × 100 = 2%.

The result is a relative error of 2%, an absolute error of 0.20, and a decimal relative error of 0.0200. The last value is rounded to four decimal places; the percentage is rounded to two.

When Relative Error Is Useful

Absolute error tells you how far apart two values are in the original unit. Relative error puts that difference in proportion to the reference, which helps make comparisons across quantities of different sizes.

For example, an error of 1 unit is substantial against a reference of 5 units, but small against a reference of 1,000 units. Their absolute errors are equal, yet their relative errors differ: 20% and 0.1%, respectively. Relative error is therefore useful when you want to judge the size of a discrepancy relative to the value being measured.

It commonly appears when checking measurements against known values, evaluating approximations, or comparing numerical results. The reference must be meaningful for the question: changing the reference changes the relative error.

How to Interpret the Result

A relative error of 0% means the measured and reference values are equal. A result of 2% means the absolute difference is 2% of the magnitude of the reference value. It does not mean the measurement is necessarily accurate enough for a particular purpose; that judgment depends on the task and its required tolerance.

The percentage alone also hides direction. If a measured value is 98 against a reference of 100, the signed error is −2%, indicating an underestimate. If it is 102, the signed error is +2%, indicating an overestimate. Both have an absolute relative error of 2%.

Common Mistakes and Limitations

• Dividing by the measured value: The standard formula uses the reference value in the denominator, not the measured value.
• Mixing units: Convert values to the same unit before subtracting. For example, do not directly compare 1 metre with 90 centimetres; convert one so both are expressed in metres or centimetres.
• Confusing ratio and percentage: A relative error of 0.02 is the same as 2%, not 0.02%.
• Rounding too early: Keep sufficient precision during the calculation and round the final result to an appropriate number of decimal places.
• Using a zero reference: If the reference is zero, the formula requires division by zero and relative error is undefined. Report the absolute difference or choose a suitable nonzero reference if the context allows.
• Ignoring measurement uncertainty: A calculated difference does not by itself show whether either value is reliable. Consider the precision and uncertainty of the measurements when interpreting the result.

Relative error can also be misleading when the reference is very close to zero: even a small absolute difference can produce a very large percentage. In that situation, report the absolute error alongside the relative value, or use another comparison measure suited to the data.

Relative Error vs. Absolute Error

Absolute error is the magnitude of the difference, |m − r|, expressed in the same units as the original values. Relative error scales that difference by the reference magnitude, so it has no unit. They answer different questions: “How many units apart are the values?” versus “How large is the difference compared with the reference?”

Neither measure is universally better. Use absolute error when the practical impact depends on the amount of difference in the original unit. Use relative error when proportional comparison matters, while remembering that its value depends on a nonzero reference and can become unstable near zero.

Frequently Asked Questions

How do you calculate relative error as a percentage?
Subtract the reference value from the measured value, take the absolute value, divide by the absolute reference value, and multiply by 100.

Can relative error be negative?
Standard relative error is nonnegative because it uses an absolute difference. Signed relative error can be negative when the measured value is below the reference; state which convention you use.

What is the difference between relative error and percent error?
Relative error is often given as a decimal ratio. Percent error is that ratio multiplied by 100. In many settings, “relative error” also refers to the percentage form, so check the requested convention.

What if the reference value is zero?
Relative error is undefined because the formula divides by the reference value. Report the absolute error or use an alternative measure appropriate to the comparison.

Does a smaller relative error always mean a better measurement?
Not necessarily. A smaller value indicates a smaller discrepancy relative to the chosen reference, but whether it is acceptable depends on the measurement’s purpose, uncertainty, and required tolerance.

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in m | Measured value | - | 9.8
in r | Reference value | - | 10
out re | Relative error | % | abs(m - r) / abs(r) * 100 | 2
out d | Absolute error | - | abs(m - r) | 2
out rd | Relative error | - | abs(m - r) / abs(r) | 4