Explain Absolute Error, Relative Error and Percentage Error.

$(a)$ Absolute Error:

The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement.

It is denoted by $|\Delta a|$.

In absence of any other method, we consider arithmetic mean as true value.

Consider physical quantity ' $a$ '. Its measurement be $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ Average value,

$\therefore a_{\text {mean }}=\frac{a_{1}+a_{2}+a_{3}+\ldots a_{n}}{n}$ OR

$\sum^{n} a_{i}$

$=\frac{i=1}{n} \text { where, } i=1,2,3, \ldots, n$

$(b)$ Absolute error in measurement

$\Delta a_{1}=a_{1}-a_{\text {mean }}$

$\Delta a_{2}=a_{2}-a_{\text {mean }}$

$: \quad: \quad:$

$\Delta a_{n}=a_{n}-a_{\text {mean }}$

$\Delta a$ may be positive or negative.

Average absolute error is denoted by $(\Delta a)_{\text {mean }}$

$=\frac{\left|\Delta a_{1}\right|+\left|\Delta a_{2}\right|+\ldots\left|\Delta a_{n}\right|}{n}$

$=\frac{\sum_{i=1}^{n}\left|\Delta a_{i}\right|}{n}$

where, $i=1,2,3, \ldots, n$

Physical quantity is represented as,

$a=a_{\text {mean }} \pm(\Delta a)_{\text {mean }}$

$\text { OR } a_{\text {mean }}-\Delta a_{\text {mean }} \leq a \leq a_{\text {mean }}+\Delta a_{\text {mean }}$