Approximation Error - Overview

Overview

One commonly distinguishes between the relative error and the absolute error. The absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. The percent error is the relative error expressed in terms of per 100.

As an example, if the exact value is 50 and the approximation is 49.9, then the absolute error is 0.1 and the relative error is 0.1/50 = 0.002. The percent error would then be 0.002*100=0.2%. Another example would be if you measured a beaker and read 5mL. The correct reading would have been 6mL. This means that your percent error would be 16.67%.

The relative error is often used to compare approximations of numbers of widely differing size; for example, approximating the number 1,000 with an absolute error of 3 is, in most applications, much worse than approximating the number 1,000,000 with an absolute error of 3; in the first case the relative error is 0.003 and in the second it is only 0.000003.

There are two features of relative error that should be kept in mind. Firstly, relative error is undefined when the true value is zero as it appears in the denominator (see below). Secondly, relative error is sensitive to the units of the true value. For example, when an absolute error in a temperature measurement given in Celsius is 1° and the true value is 2°C, the relative error is 0.5 and the percent error is 50%. For this same case, when the temperature is given in Kelvin, the same 1° absolute error with the same true value of 275.15° K gives a relative error of 3.63e-3 and a percent error of only 0.363%. Clearly one could manipulate the relative error by choosing units to make this relative error look either very large or very small. Absolute error, which is given in the units of the two values, does not share these problems; in this example, the absolute error is either 1°C or 1° K, and would be defined even if the true value was zero.