The approximation error in some data is the discrepancy between an exact value and some approximation to it. An approximation error can occur because

1. the measurement of the data is not precise (due to the instruments), or
2. approximations are used instead of the real data (e.g., 3.14 instead of π).

In the mathematical field of numerical analysis, the numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm.

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 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 .003 and in the second it is only .000003.

Definitions

Given some value v and its approximation v_approx, the absolute error is

\epsilon = |v_{\text{approx}} - v|\,

where the vertical bars denote the absolute value. If v\ne 0, the relative error is

\eta = \frac{|v_{\text{approx}}-v|}{|v|},

and the percent error is

\delta = \frac{|v_{\text{approx}}-v|}{|v|}\times{}100.

These definitions can be extended to the case when v and v_{\text{approx}} are n-dimensional vectors, by replacing the absolute value with a 2-norm^[1].

Compared with absolute error, relative error is more meaningful as a measure of accuracy. Sometimes, absolute error may mislead us in considering the accuracy of an appoximate value. Let's see two examples. Suppose v_{\text{approx}} is an appoximation to value v,

Example a) We have v=0.5\times10^1 and v_{\text{approx}}=0.51\times10^1, the absolute error is 0.1 and the relative error is 0.02.

Example b) We have v=0.5000\times10^4and v_{\text{approx}}=0.5100\times10^4, the absolute error is 100 and the relative error is 0.02.

The absolute error in example b) seems very large and unacceptable, but it has the same relative error as that in example a). So the absolute error in example b) is misleading.

See also

• Uncertainty
• Propagation of uncertainty

References

External links

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