Statistics formulas





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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