Percent Error Calculator

Enter what you measured and what the value should be. With the defaults, an observed value of 10 against a true value of 11, the error is 1 and the percent error is about 9.09%.

Percentage Error

Percentage error, also called percent error, expresses how far a measured or estimated value is from the true value as a percentage of the true value. It is used in science classes and laboratories to judge an experiment, in engineering to check a model against reality and in forecasting to grade predictions. Expressing the error as a percentage puts results of different sizes on an equal footing: being off by 1 gram matters a lot when weighing 5 grams and very little when weighing 5 kilograms.

Error is different from mistake. Every measurement has some error, from the limited resolution of instruments, from random variation and from systematic effects such as a scale that is not zeroed. Percent error lumps these together into a single comparison with a known answer, so it is only possible when a trusted reference value exists, such as the accepted density of water or the value printed on a calibration weight.

What counts as a good percent error depends on the field. A classroom experiment measuring gravitational acceleration might be satisfied with under 5%, while precision manufacturing may demand a fraction of a percent. Percent error is closely related to percent difference, which compares two measurements when neither is known to be correct and divides by their average instead of by a true value.

Computing percentage error

Subtract the true value from the observed value, take the absolute value, divide by the absolute value of the true value, and multiply by 100. Suppose a student measures the boiling point of water at sea level as 97.5 °C, while the accepted value is 100 °C.

• Find the difference: 97.5 − 100 = −2.5.
• Take the absolute value: |−2.5| = 2.5.
• Divide by the true value: 2.5 ÷ 100 = 0.025.
• Multiply by 100: 0.025 × 100 = 2.5%.

Percent error = |Vobserved − Vtrue| ÷ |Vtrue| × 100%

Vobserved

the measured, estimated or experimental value

Vtrue

the true, accepted or theoretical value

Negative percentage error

The standard formula uses absolute values, so the percent error is never negative; it reports only the size of the miss. In many contexts the direction matters too, and the signed percent error drops the absolute value from the numerator: (observed − true) ÷ |true| × 100%. A negative result means the measurement came in below the true value, and a positive result means it came in above. The boiling point example has a signed error of −2.5%, an underestimate.

The sign is useful for spotting systematic error. If repeated measurements all have negative errors, the instrument or method is probably biased low, for example a thermometer that reads consistently cold or a ruler with a worn end. Random errors, by contrast, scatter both above and below zero. When reporting results, state which convention you are using, since a signed error of −9.09% and an absolute error of 9.09% describe the same measurement.

One caution: percent error is undefined when the true value is zero, and it can be misleading when the true value is very small, because tiny absolute differences turn into huge percentages. In those cases, report the absolute error instead.

Frequently asked questions