and hence . » f ( )( . ) 03333 ′ Now if we use a calculator, just to compare, we find 03315 . We see that our approximation is accurate to three decimal places and the error is 03315 03333 00018 = − .
[Also note that one could choose f x x x and ∆ x = . . So the choice of f and x are not necessarily unique]. So in the above example, the absolute error is 03315 03333 .00018.
. Note that the absolute error says how much the error; but it does not say how good the approximation is. For instance, let us consider two simple cases. Case : Suppose that the actual value of something is and its approximated value is , then the absolute error is = .
Case : Suppose that the actual value of something is and its approximated value is . In this case, the absolute error is . So the absolute error in the first case is smaller when compared to the second case. Among these two approximations, which is a better approximation; and why?
The absolute error does not give a clear picture about whether an approximation is a good one or not. On the other hand, if we calculate relative error or percentage of error (defined below), it will be easy to see how good an approximation is. If the actual value is zero, then we do know how close our approximate answer is to the actual value. So if the actual value is not zero, then we define, and - - Differentials and Partial Derivatives Definition .
If the actual value is not zero, then Relative error = Actual value Approximate value Actual value | | Percentage error = Relative error × Note : Absolute error has unit of measurement where as relative error and percentage error are units