# Definition:Error

## Definition

Let $x_0$ be an approximation to a (true) value $x$.

The error $\Delta x$ is an indicator of how much difference there is between $x$ and $x_0$.

### Absolute Error

The absolute error of $x_0$ in $x$ is defined as:

$\Delta x := x_0 - x$

### Relative Error

The relative error of $x_0$ in $x$ is defined as:

$\delta x := \dfrac {\Delta x} x$

where $\Delta x$ denotes the absolute error of $x_0$.

This can be defined only when $x \ne 0$.

### Percentage Error

The percentage error of $x_0$ in $x$ is defined as the relative error expressed as a percentage:

$\delta x \, \% := \delta x \times 100$

This, like the relative error, can be defined only when $x \ne 0$.

## Also defined as

### Absolute Error

The absolute error of $x_0$ in $x$ can also be seen defined as:

 $\text {(1)}: \quad$ $\ds \Delta x$ $:=$ $\ds x - x_0$ $\text {(2)}: \quad$ $\ds \Delta x$ $:=$ $\ds \size {x_0 - x}$

where $\size {x_0 - x}$ denotes the absolute value of $x_0 - x$.

### Relative Error

The relative error of $x_0$ in $x$ can also be defined as:

$\delta x \approx \dfrac {\Delta x} {x_0}$

where:

$\Delta x$ denotes the absolute error of $x_0$
$\approx$ indicates that the value is but approximate.

This can be particularly useful when the true value $x$ can only be speculated.

## Also known as

An error is also, in some branches of mathematics, known as a residual.

## Also see

• Results about errors can be found here.

Not to be confused with a mistake.