Relative Error Calculator
Enter a measured (observed) value and the actual (true) value to find the absolute error, relative error as a decimal, and percent error. You can also reverse-solve: provide any two of the three quantities (measured value, actual value, or percent error) and the calculator finds the third. Results update as you type.
Formula
Worked example
A student measures gravitational acceleration as 9.8 m/s2. The accepted value is 9.81 m/s2. Absolute error = |9.81 - 9.8| = 0.01 m/s2. Relative error = 0.01 / 9.81 = 0.00102. Percent error = 0.102%. This falls in the excellent range.
What is relative error?
Relative error expresses how large an error is in proportion to the true (reference) value. A raw difference of 5 metres means something very different when measuring a house versus measuring the distance to the Moon, so dividing by the actual value produces a dimensionless ratio that is comparable across different scales and units. Multiplied by 100, it becomes the familiar percent error. For example, a student who measures a room as 4.95 m when it is actually 5.00 m has an absolute error of 0.05 m and a relative error of 0.05 / 5.00 = 0.01, or 1%. Two common conventions exist: some fields define relative error as a signed quantity (retaining direction of bias), while others take the absolute value to get a magnitude only. This calculator uses the absolute-value convention, which is standard in most laboratory and engineering settings.
Absolute error vs. relative error vs. percent error
Absolute error is simply |actual - measured|, the unsigned gap in the original units. It tells you the raw magnitude of the mistake but not how significant that mistake is relative to the size of the thing being measured. Relative error fixes this by dividing the absolute error by the actual value, yielding a dimensionless ratio. Percent error is just relative error multiplied by 100 so it reads as a familiar percentage. All three describe the same error from different perspectives: if you weigh a 200 g object and record 195 g, the absolute error is 5 g, the relative error is 0.025, and the percent error is 2.5%.
How to use the reverse-solve mode
Sometimes you know the error tolerance and one of the two values, and you need to find the other. Use the 'Solve for' drop-down to switch modes. In 'Measured Value' mode, enter the true reference value and the allowed percent error: the calculator shows the measured value (or values) that would produce exactly that error. In 'Actual Value' mode, enter your observed measurement and the known percent error to back-calculate the implied actual value. Note that because relative error uses an absolute value, each reverse-solve has two algebraic solutions (one above and one below the reference). Both are shown in the steps panel.
Common sources of relative error in practice
Relative error arises from two broad categories. Random error is the unpredictable scatter in repeated measurements caused by things like human reaction time, electrical noise, or environmental fluctuations; it averages toward zero over many trials and is reduced by taking more readings. Systematic error is a consistent bias in one direction caused by a mis-calibrated instrument, a rounding in a physical constant, or a flawed measurement technique; it does not average away and must be corrected at the source. A high percent error that persists across repeated measurements is a strong sign of systematic error. Comparing relative errors across instruments or laboratories is the standard way to identify which method is more accurate.
Relative error severity reference
| Percent error | Severity | Interpretation |
|---|---|---|
| 0 - 1% | Excellent | Instrument or method is highly precise; suitable for critical measurements |
| 1 - 5% | Acceptable | Adequate for most engineering and applied science work |
| 5 - 10% | Moderate | Review measurement procedure; may be acceptable for preliminary estimates |
| 10 - 25% | High | Investigate for systematic error or instrument drift |
| Above 25% | Very high | Result is unreliable; check reference value, units, and method |
General guidelines used across scientific, engineering, and laboratory contexts. Acceptable thresholds vary by discipline.
Frequently asked questions
What is the formula for relative error?
Relative error = |actual value - measured value| / |actual value|. To express it as a percent error, multiply by 100. The vertical bars denote absolute value, so the result is always zero or positive regardless of whether the measurement is above or below the true value.
Can relative error be negative?
Not with the standard absolute-value definition used here. By taking |actual - measured|, the formula discards the sign. Some disciplines define a signed relative error (measured - actual) / actual to preserve direction of bias, but that is less common and the result is called signed or directional relative error to distinguish it.
Can relative error be greater than 100%?
Yes. If the measured value is more than twice the actual value (or very close to zero while the actual is large), the formula yields a result above 1, meaning above 100%. For example, if the actual value is 2 and the measured value is 7, the percent error is |2 - 7| / 2 x 100 = 250%.
What is the difference between percent error and percentage difference?
Percent error compares a single measurement to a known or accepted reference value using that reference as the denominator. Percentage difference compares two measured values when neither is considered the 'true' value, using their average as the denominator: |A - B| / ((A + B) / 2) x 100. Use percent error when you have a reference; use percentage difference when comparing two measurements of equal standing.
What percent error is acceptable?
Acceptable thresholds depend entirely on context. High-precision physics or metrology work may require under 0.1%, while a field survey or preliminary chemistry experiment may accept 5-10%. The reference table on this page gives common conventions. When a standard is not specified, error under 5% is generally considered acceptable and error under 1% is considered excellent.
Why can I not calculate relative error when the actual value is zero?
Relative error divides by the actual value, so if the actual value is zero, the formula involves division by zero, which is mathematically undefined. In situations where the true value is zero or very close to it, absolute error or the closely related absolute relative error with a different normalizing denominator (such as the measurement itself) is used instead.