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Approximation Error

At times in a statistical measurement, it is important to determine an approximation error. In a simpler form, let there be some quantity $x$ that is being applied some operation or measurement in order to obtain another quantity $y$. If a small error is produced in $x$, then eventually $y$ = $f(x)$ will also reflect an error. It is said to be a discrepancy between approximated value and exact value. In this article, we are going to learn about the approximation error, relative approximation error and other important concepts related to it.

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Definition

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A deviation between the exact value of a data and its approximated value is referred to an approximation error. This error is produced while measurement or calculation. There may be following two main causes for approximation error to occur :

(1) The measuring instrument is not precise in order to accept tiny values. For example - A ruler may be accurate up to one decimal point, $4.6$, $8.3$ etc,  but it is unable to measure up to two places of decimals.

(2) When instead to exact value, approximations are used in calculations, such as taking $3.14$ for $\pi$.

Let there be some value x and $\widetilde{x}$ be its approximated value, then approximated error can also be measured by -
$\epsilon = | x - \widetilde{x}|$

where, $\epsilon$ is used to denote approximated error.

An approximated error can also be known as an absolute error because the absolute value of the error is considered.

Relative Approximation Error

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Relative approximation error is represented by the ratio of absolute approximation error to the absolute value of given value. If relative approximation error is denoted by n, then

$n$ = $\frac{\epsilon}{|x|}$

$n$ = $\frac{| x\ -\ \widetilde{x}|}{|x|}$

$n$ = $|\frac{ x\ -\ \widetilde{x}}{x}|$

$n$ = $|1\ -\ \frac{\widetilde{x}}{x}|$

It is also known as just "relative error".

The percentage of the relative error is referred as the percentage error.

Percentage
error = $|1\ -\ \frac{\widetilde{x}}{x}|$ $ \times\  100$ %

Norm of The Approximation Error

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Above definitions of approximation error and relative approximation error consider one-dimensional vectors $x$ and $\widetilde{x}$. When these vectors are defined in n dimensions, the absolute value is to be replaced by an n-norm.

A norm $p$ can be defined in the following way. Let p be a norm and V be the given vector space, then $p\ :\ V$ $\rightarrow$ $R$ is said to be a vector $x$ $\in$ $V$ which is represented in the form of enclosing $x$ within two pairs of vertical lines, i.e. $\left \| x \right \|$ = $p(x)$.

Examples

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Let's have a look at few examples discussed below.
Example 1: If exact value of a variable is $50$. But in an experiment, it is approximated $49.9$. Find the absolute approximation error and relative error.

Solution: Here, $x$ = $50$ and $\widetilde{x}$ = $49.9$

$\epsilon$ = $| x\ -\ \widetilde{x}|$

$\epsilon$ = $|\ 50\ -\ 49.9|$

$\epsilon$ = $0.1$

Thus, absolute error is $0.1$.

Relative error is given by :

$n$ = $\frac{\epsilon}{|x|}$

= $\frac{0.1}{50}$ = $0.002$

Percentage error = $0.002\ \times\ 100$% = $0.2$
Example 2: A quantity in a beaker should exactly be $6$ ml. But the approximated value came out to be $5$ ml. Calculate the relative error.

Solution: Here, $x$ = $6$ ml and $\widetilde{x}$ = $5$ ml

$\epsilon$ = $| x\ -\ \widetilde{x}|$

$\epsilon$ = $| 6\ -\ 5|$

$\epsilon$ = $1$

Therefore, absolute error is $1$.

Relative error is calculated as under :

$n$ = $\frac{\epsilon}{|x|}$

= $\frac{1}{6}$ = $0.1667$

Percentage error = $0.1667\ \times\ x\ 100$% = $16.67$%.
Example 3 : The exact area of a square is $240$ sq m. The approximated area is measured to be $240.6$ sq m. Determine the relative and percentage error.

Solution : Here, $x$ = $240$ sq m and $\widetilde{x}$ = $240.6$ sq m

$\epsilon$ = $| x\ -\ \widetilde{x}|$

$\epsilon$ = $|240\ -\ 240.6|$

$\epsilon$ = $0.6$

The absolute error is $0.6$

Relative error :

$n$ = $\frac{\epsilon}{|x|}$

= $\frac{0.6}{240}$ = $0.0025$

Percentage error = $0.0025\ \times\ 100$% = $0.25$%.
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