Percentage error is a measurement of the discrepancy between an observed (measured) and a true (expected,
accepted, known etc.) value. It is typically used to compare measured vs. known values as well as to
assess whether the measurements taken are valid.
When measuring data, whether it be the density of some material, standard acceleration due to gravity of
a falling object, or something else entirely, the measured value often varies from the true value. Error
can arise due to many different reasons that are often related to human error, but can also be due to
estimations and limitations of devices used in measurement. Calculating the percentage error provides a
means to quantify the degree by which a measured value varies relative to the true value. A small
percentage error means that the observed and true value are close while a large percentage error
indicates that the observed and true value vary greatly. In most cases, a small percentage error is
desirable, while a large percentage error may indicate an error or that an experiment or measurement
technique may need to be re-evaluated. If, for example, the measured value varies from the expected
value by 90%, there is likely an error, or the method of measurement may not be accurate.
Computing percentage error
The computation of percentage error involves the use of the absolute error, which is simply the
difference between the observed and the true value. The absolute error is then divided by the true
value, resulting in the relative error, which is multiplied by 100 to obtain the percentage error. Refer
to the equations below for clarification.
Absolute error = |V_{observed} – V_{true}|
Relative error =
|V_{observed} – V_{true}|
V_{true}
Percentage error =
|V_{observed} – V_{true}|
V_{true}
× 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 equations above are based on the assumption that true values are known. True values are often
unknown, and under these situations, standard deviation is one way to represent the error. Please refer
to the standard deviation calculator for further
details.
Negative percentage error
Based on the formula above, when the true value is positive, percentage error is always positive due to
the absolute value. In most cases, only the error is important, and not the direction of the error.
However, it is possible to have a negative percentage error. This occurs if we do not take the absolute
value of the error, the observed value is smaller than the true value, and the true value is positive.
For example, given an observed value of 7, a true value of 9, and allowing for a negative percentage,
the percentage error is:
v_{observed} – v_{true}
v_{true}
× 100% =
7 – 9
9
× 100%
=
-22.222%
A negative percentage error simply means that the observed value is smaller than the true value. If the
observed value is larger than the true value, the percentage error will be positive. Thus, in the
context of an experiment, a negative percentage error just means that the measured value is smaller than
expected. It does not indicate that the observed value is somehow better than expected, since the best
possible outcome for percentage error is that the observed and true values are equal, resulting in a
percentage error of 0.