# Statistics:

Statistics is a branch of mathematics which deals with the study of collection , organization and interpretation of data.

## a>Arithmetic mean formulas:

1> The mean for ungroup ( individual ) data or the arithmetic mean is denoted by: $\bar{X}$ and it’s formula is: $\bar{X} = \dfrac{\displaystyle\sum X}{N}$

2>The arithmetic mean for group (discrete) data is calculated using formula: $\bar{X} = \dfrac{\displaystyle\sum fX}{N}$

3> The arithmetic mean for continuous data is calculated using the formulas:

Direct method:

$\bar{X} = \dfrac{\displaystyle\sum fX}{f}$

Deviation method:

$\bar{X} =A + \dfrac{\displaystyle\sum fd}{f}$

Step deviation method:

$\bar{X} =A + \dfrac{\displaystyle\sum fd}{f} \times i$

Where , d = X – A , A = assumed mean and i = height of the class.

## b> Median formulas:

1> Median for ungroup data:

i> Median = $\dfrac{N+1}{2}$‘th observation if the “N” is odd.

ii> Median = average of $\left(\dfrac{N}{2}\right)^{th}$ and $\left(\dfrac{N}{2} +1 \right)^{th}$ observation if “N” is even.

2> Median for group data:

Median = $l + \dfrac{\frac{N}{2} - c.f.}{f} \times h$ ( For continuous frequency distribution )

where , l= lower limit of median class , f= frequency of median class , c.f. = cumulative frequency of pre-median class , h= size of median class , N = total numbers of items.

## c> Quartiles formulas :

1> First quartile:

For individual data:

$Q_1 = \left( \dfrac{n+1}{4}\right)^{th}$ observation in the data arranged in ascending or decreasing order.

For group ( discrete) data:

$Q_1 = \left( \dfrac{n+1}{4}\right)^{th}$ observation in the data arranged in ascending or decreasing order.

For continuous data:

$q_1$ lies in $\left(\dfrac{N}{4}\right)^{th}$place.

and exact first quartile = $l + \dfrac{\frac{N}{4} - c.f.}{f} \times h$

where , , l= lower limit of first quartile class , f= frequency of first quartile class , c.f. = cumulative frequency of pre-first quartile class , h= size of first quartileclass , N = total numbers of items.

2> Third quartile:

For individual data:

$Q_3 = \left( \dfrac{3(n+1)}{4}\right)^{th}$ observation in the data arranged in ascending or decreasing order.

For group ( discrete) data:

$Q_3 = \left( \dfrac{3(n+1)}{4}\right)^{th}$ observation in the data arranged in ascending or decreasing order.

For continuous data:

$Q_3$ lies in $\left(\dfrac{3N}{4}\right)^{th}$place.

and exact third quartile = $l + \dfrac{\frac{3N}{4} - c.f.}{f} \times h$

where , , l= lower limit of third quartile class , f= frequency of third quartile class , c.f. = cumulative frequency of pre-third quartile class , h= size of third quartileclass , N = total numbers of items.

## d> Mode formulas:

1> Mode = value with highest frequency for ungroup and discrete frequency distribution.

2> Mode ($M_o$) = $l + \dfrac{f_1-f_0}{2f_1 - f_0 - f_2} \times h$ ( for continuous frequency distribution)

3> If $f_0 > f_1$ or $f_2 > f_1$ then,

Mode = $l +\dfrac{f_2}{f_0 - f_2} \times h$

4> If $2f_1 - f_0 - f_2 = 0$ then ,

Mode = $\dfrac{f_1-f_0}{ | f_1 - f_0 | + | f_1 - f_2 | } \times h$

5> If two or more than two highest frequency data are present then ,

Mode = $3 \times median - 2 \times mean$

Where , l = lower limit of mode class , $f_1$ = frequency of modal class , $f_0$ = frequency of the class preceding modal class , $f_2$ = the frequency of class succeeding modal class , h = class width or height

## e> Deviation formulas:

1> Mean deviation:

Mean deviation from mean ( $M.D.$ )

For ungroup data:

$M.D. = \dfrac{\sum d}{N}$

For group ( continuous and discrete ) data:

$M.D. = \dfrac{\sum f d}{f}$ and $d = | X - \bar{X} |$

Mean deviation from mode ( $M.D. _{mo}$ )

For ungroup data:

$M.D. _{mo} = \dfrac{\sum d}{N}$ and $d = | X - \bar{X} |$

For group ( continuous and discrete ) data:

$M.D. _{mo} = \dfrac{\sum fd}{f}$ where , $d = | X - \bar{M_o} |$

2> Quartile deviation:

Inter quartile range = $Q_3 - Q_1$

Quartile deviation or semi – Inter quartile range = $Q.D = \dfrac{Q_3 - Q_1}{2}$

Co-efficient of quartile deviation = $Q.D = \dfrac{Q_3 - Q_1}{Q_3 + Q_1}$

4> Standard deviation:

For ungroup data:

Standard deviation ( $\sigma$ ) = $\sqrt{\dfrac{\sum d^3}{N}}$ where , $d = | X - \bar{X} |$ ( $\bar{X}$ is assumed mean).

For group data i :

Standard deviation ( $\sigma$ ) = $\sqrt{\dfrac{\sum d^3}{f}}$ where , $d = | X - \bar{X} |$ ( $\bar{X}$ is actual mean).

For group data ii :

Standard deviation ( $\sigma$ ) = $\sqrt{ \dfrac{ \sum fd^2 }{ N } - \left( \dfrac{ \sum fd}{N}\right)^2}$
where , $d = | X - \bar{X} |$ ( $\bar{X}$ is assumed mean).

5> Range:

Range = Largest item – smallest item = $L - S$

Co-efficient of range = $\dfrac{L - S}{L +S}$

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