Error (of an approximation)
Elementary
Think of an estimate or a measurement of some quantity, like the length of a table. The error of the estimate is how far it is from the true value.
High school
Consider an approximate or measured value $\hat{x}$ of a true value $x$. The error of $\hat{x}$ is how far it is from $x$. The absolute error is $\abs{\hat{x} - x}$, and the relative error is $\abs{\hat{x} - x} / \abs{x}$.
Examples
Examples
-
$9.8 - 9.81 = -0.01$, for a measured value $9.8$ and true value $9.81$
Elementary
$9.8 - 9.81 = -0.01$ is the (signed) error of the measurement, because it compares the measured value $9.8$ with the true value $9.81$.
-
In floating point arithmetic, $0.1 + 0.2$ evaluates to $0.30000000000000004$ due to rounding errors.
High school
The extra $0.00000000000000004$ is an error: the rounding error of floating-point arithmetic.
Notes
Sign conventions vary: some authors define the error as $\hat{x} - x$, and others as $x - \hat{x}$.
Distinctions involving this term
- Types of difference: Many quantities compare two things by “how far apart” they are. A difference is the result of subtraction and can be negative. No distance is negative. An error compares an approximation with the truth, a change compares the end with the start, and a set difference removes elements.