Statistics

Mean of un-grouped data:

If ${\text{x}}_{\text{1}}{\text{, x}}_{\text{2}}{\text{, x}}_{\text{3}}\text{, }\mathrm{...}{\text{ x}}_{\text{n}}$ are observations with frequencies ${\text{f}}_{\text{1}}{\text{, f}}_{\text{2}}{\text{, f}}_{\text{3}}\text{, }\mathrm{...}{\text{ f}}_{\text{n}}$ then

 

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

 

$\text{= }\frac{{\displaystyle \sum {\text{f}}_{\text{i}}{\text{x}}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}$

 

Mean of Grouped data:

 

Class mark for a class:

Class mark $\text{= }\frac{\text{uppar limit + lower limit}}{\text{2}}$

 

 

Mean of grouped data:

1)      Direct method:

 

If ${\text{x}}_{\text{i}}\text{'}$ s are class marks of the classes with frequencies ${\text{f}}_{\text{i}}$

 

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

 

$\overline{\text{x}}\text{ = }\frac{{\displaystyle \sum {\text{f}}_{\text{i}}{\text{x}}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}$

 

 

e.g.

 

 

Mean $\text{= }\overline{\text{x}}\text{ = }\frac{{\displaystyle \sum {\text{f}}_{\text{i}}{\text{x}}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}\text{ = }\frac{\text{128}}{\text{12}}\text{ = 10}\text{.66}$

 

 

 

2) Assumed mean method:

Mean $\text{= }\frac{{\displaystyle \sum {\text{f}}_{\text{i}}{\text{d}}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}\text{ + A}$, where $\text{A}$ is assumed mean.

 

e.g. For following data assumed mean is 12

 

Mean $\text{= }\overline{\text{x}}\text{ = A + }\frac{{\displaystyle \sum {\text{f}}_{\text{i}}\text{d}{}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}\text{ + A = 12 + }\left(\frac{\text{-16}}{\text{12}}\right)\text{ = 12 + }\left(\text{-1}\text{.33}\right)\text{ = 10}\text{.66}$

 

 

2)      Step deviation method:

 

$\overline{\text{x}}\text{ = A + h}\frac{{\displaystyle \sum {\text{f}}_{\text{i}}{\text{u}}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}$

 

For e.g.

If Assumed mean $\text{= 12}$ , $\text{h =}$ length of class interval $\text{= 4}$

 

 

 

Mean $\text{= }\overline{\text{x}}\text{ = A + }\frac{{\displaystyle \sum {\text{f}}_{\text{i}}\text{u}{}_{\text{i}}}}{{\displaystyle \sum {\text{f}}_{\text{i}}}}\text{ + A = 12 + 4}\left(\frac{\text{-4}}{\text{12}}\right)\text{ = 12 + 4}\left(\text{-0}\text{.33}\right)\text{ = 10}\text{.66}$

 

 

Median:

Median of ungrouped data:

When data is in ascending order,

 

If n = total number of observations

 

Median $\text{= }\{\begin{array}{l}{\left(\frac{\text{n + 1}}{\text{2}}\right)}^{\text{th}}\text{observation}\mathrm{...}\text{n is odd}\\ \frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{observation + }{\left(\frac{\text{n + 2}}{\text{2}}\right)}^{\text{th}}\text{observation}}{\text{2}}\mathrm{...}\text{n is even}\end{array}$

 

 

 

Median of grouped data:

Median = $\text{l + }\frac{\frac{\text{n}}{\text{2}}\text{ - cf}}{\text{f}}\text{ × h}$

 

$\text{L =}$ lower limit of median class

 

$\text{N =}$ total no. of observations

 

$\text{Cf =}$ cumulative frequency of class previous to median class

 

$\text{f =}$ frequency of median class

 

$\text{h =}$ class size

 

 

For eg.

 

 

$\text{N = 12}$

 

$\frac{\text{n}}{\text{2}}\text{ = 6}$

 

median class $\text{= 10 - 14}$

 

$\text{L =}$ lower limit of median class $\text{= 10}$

 

$\text{Cf =}$ cumulative frequency of class previous to median class $\text{= 5}$

 

$\text{f =}$ frequency of median class $\text{= 4}$

 

$\text{h =}$ class size $\text{= 4}$

 

Median $\text{= l + }\frac{\frac{\text{n}}{\text{2}}\text{ - cf}}{\text{f}}\text{ × h}$

 

$\text{= 10 + }\frac{\text{6 - 5}}{\text{4}}\text{ × 4}$

 

$\text{= 10 + 1}$

 

$\text{= 11}$

 

 

Mode:

Mode of ungrouped data:

It is most frequently occurred value in the data.

 

 

Mode of grouped data:

Mode $\text{= l + }\frac{{\text{f}}_{\text{1}}{\text{- f}}_{\text{0}}}{{\text{2f}}_{\text{1}}{\text{ - f}}_{\text{0}}{\text{- f}}_{\text{2}}}\text{ × h}$

 

$\text{L =}$ lower limit of modal class

$\text{h =}$ size of class interval

${\text{f}}_{\text{1}}\text{ =}$ frequency of modal class

${\text{f}}_{\text{0}}\text{ =}$ frequency of class preceding to modal class

${\text{f}}_2\text{ =}$ frequency of succeeding to modal class

 

 

 

 

For e.g.

 

 

Mode $\text{= l + }\frac{{\text{f}}_{\text{1}}{\text{- f}}_{\text{0}}}{{\text{2f}}_{\text{1}}{\text{ - f}}_{\text{0}}{\text{- f}}_{\text{2}}}\text{ × h}$

 

$\text{L =}$ lower limit of modal class $\text{= 10}$

$\text{h =}$ size of class interval $\text{= 4}$

${\text{f}}_{\text{1}}\text{ =}$ frequency of modal class $\text{= 4}$

${\text{f}}_{\text{0}}\text{ =}$ frequency of class preceding modal class $\text{= 3}$

${\text{f}}_2\text{ =}$ frequency of succeeding modal class $\text{= 3}$

 

 

Mode $\text{= 10 + }\frac{\text{4 - 3}}{\text{2 × 4 - 3 - 3}}\text{ × 4}$

$\text{= 10 + }\frac{\text{1}}{\text{2}}\text{ × 4}$

$\text{= 12}$

 

Relation between mean median and mode:

$\text{3}$ median $\text{=}$ mode $\text{+}$ $\text{2}$ mean

 

Ogives(cumulative frequency curves):

Ogiveis graphical representation of cumulative frequency

Two types of cumulative frequency are:

1)      cumulative frequency of less than type

2)      cumulative frequency of more than type

 

 

1) cumulative frequency curve of less than type and its curve:

 

 

 

 

2) Cumulative frequency curve of more than type and its curve:

 

 

 

 

To find median graphically:

Step 1) Draw a cumulative frequency curve of less than type and greater than type.

Step 2) $\text{x}$ coordinate of intersection of curves is the median.