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.

Sources

• 1964: B. Noble: Numerical Methods: Volume $\text { 1 }$ ... (previous) ... (next): Chapter $\text I$: Accuracy and Error: $\S 1.1$. Introduction
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): error: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): error
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): error