The percentage error quantifies how far a measured (observed) value deviates from the actual (expected, accepted, or known) value. It’s commonly employed to contrast experimental results with reference values and to judge the reliability of the measurements.
When you record data—whether it’s material density, standard gravitational acceleration of a falling body, or any other quantity—the result often differs from the true value. Such deviations stem from many sources: human mistakes, approximations, and the limitations of the measuring instruments. Computing the percentage error gives a way to express how much the observed figure strays from the true one. A low percentage error indicates the two values are close, whereas a high percentage error signals a large gap. Generally, a small error is preferred; a large one may suggest a flaw in the experiment or the measurement method. For instance, if your result is 90 % away from the expected value, the procedure is likely flawed or inaccurate.
Computing percentage error
To calculate percentage error you first find the absolute error—the straight‑line difference between the observed and true values. Divide this absolute error by the true value to get the relative error, then multiply by 100 to express it as a percentage. See the formulas below for details.
Absolute error = |Vobserved – Vtrue|
Relative error =
|Vobserved – Vtrue|
Vtrue
Percentage error =
|Vobserved – Vtrue|
Vtrue
× 100%
For example, if the observed value is 56.891 and the true value is 62.327, the percentage error is:
|56.891 – 62.327|
62.327
× 100% = 8.722%
The formulas above assume the true value is known. In practice the true value is often unknown, and in such cases the standard deviation serves as an alternative way to represent error. Consult the standard deviation calculator for more information.
Negative percentage error
According to the formula, when the reference value is positive the percentage error comes out positive because we use the absolute value. Usually only the magnitude of the error matters, not its sign. Still, a negative percentage error can appear if you omit the absolute value, the measured value is lower than the true one, and the true value is positive. For example, with an observed value of 7 and a true value of 9, the percentage error (allowing a negative sign) would be:
vobserved – vtrue
vtrue
× 100% =
7 – 9
9
× 100%
=
-22.222%
A negative percentage error simply indicates that the measured figure is lower than the true value. If the measurement exceeds the true value, the percentage error turns positive. Thus, in an experiment a negative error just means the result fell short of expectations—it doesn’t imply a better outcome. The ideal case is a zero percent error, where observed and true values match exactly.
