Formula of Arithmetic Mean, Median, Mode and Properties of Arithmetic Mean

Arithmetic Mean 

It is the sum of observation divided by number of observation. It is denoted by $\overline{X}$.

Formula of Arithmetic Mean 

For n terms,

$\overline{X}=\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+...+{{x}_{n}}}{n}$

$\overline{X}=\frac{1}{2}(\sum\limits_{i=1}^{n}{{{x}_{i}}}),here\sum\limits_{i=1}^{n}{{{x}_{i}}}$   denotes   ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}+..+{{x}_{n}}$

For n terms with frequency of each term given,

$ \overline{X}=\frac{{{f}_{1}}{{x}_{1}}+{{f}_{2}}{{x}_{2}}+...+{{f}_{n}}{{x}_{n}}}{{{f}_{1}}+{{f}_{2}}+{{f}_{3}}+...{{f}_{n}}}$

$\overline{X}=\frac{\sum\limits_{i=1}^{n}{{{f}_{i}}}{{x}_{i}}}{N}$

Where  $N={{f}_{1}}+{{f}_{2}}+...+{{f}_{n}}$

Median

Formula of Median

For n terms,

If n = odd, then

Median = Value of  ${{(\frac{n+1}{2})}^{th}}$  observation

If n = even, then

Median =   $\frac{valueof{{(\frac{n}{2})}^{th}}observation+valueof{{(\frac{n}{2}+1)}^{th}}observation}{2}$

Mode 

It is the value which occurs most frequently in a set of observations.

Properties of Arithmetic Mean

1. If $\overline{X}$    is the mean of n observations ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$   , then prove that $\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{X})=0.}$  i.e. the algebraic sum of deviations from mean is zero.

2. If $\overline{X}$    is the mean of n observations ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$   , then the mean of the observations ${{x}_{1}}+a,{{x}_{2}}+a,...,{{x}_{n}}+a$   is $\overline{X}+a.$   i.e. if each observation is increased by a, then the mean is also increased by a.

3. If $\overline{X}$    is the mean of n observations ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$   , then the mean of $a{{x}_{1,}}a{{x}_{2,}}a{{x}_{3,}}...,a{{x}_{n}}$   is $a\overline{X},$   where a is any number different from zero i.e. if each observation is multiplied by a non-zero number a, then the mean is also multiplied by a.

4. If $\overline{X}$    is the mean of n observations ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$   , then the mean of $\frac{{{x}_{1}}}{a},\frac{{{x}_{2}}}{a},\frac{{{x}_{3}}}{a},...,\frac{{{x}_{n}}}{a}$    is $\frac{\overline{X}}{a},$    where a is any non-zero number. I.e. if each observation is divided by a non-zero number, then the mean is also divided by it.

5. If $\overline{X}$    is the mean of n observations ${{x}_{1}},{{x}_{2}},..,{{x}_{n}}$   , then the mean of ${{x}_{1}}-a,{{x}_{2}}-a,...,{{x}_{n}}-a$    is $\overline{X}-a,$   where a is any real number.



 Click Home for more Formulas and Properties