# Statistics

## 14 STATISTICS

Mean of un-grouped data:

If are observations with frequencies then

Mean

Mean of Grouped data:

Class mark for a class:

Class mark

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

e.g.

 Class Frequencies $\left({\text{f}}_{\text{i}}\right)$ Class mark $\left({\text{x}}_{\text{i}}\right)$ ${\text{f}}_{\text{i}}{\text{x}}_{\text{i}}$ 2-6 2 4 8 6-10 3 8 24 10-14 4 12 48 14-18 3 16 48

Mean

2) Assumed mean method:

Mean , where $\text{A}$ is assumed mean.

e.g. For following data assumed mean is 12

 Class Frequencies $\left({\text{f}}_{\text{i}}\right)$ Class mark $\left({\text{x}}_{\text{i}}\right)$ ${\text{f}}_{\text{i}}{\text{d}}_{\text{i}}$ 2-6 2 4 -8 - 16 6-10 3 8 -4 -12 10-14 4 12 - A 0 0 14-18 3 16 4 12

Mean

2)      Step deviation method:

For e.g.

If Assumed mean , length of class interval

 Class Frequencies $\left({\text{f}}_{\text{i}}\right)$ Class mark $\left({\text{x}}_{\text{i}}\right)$ ${\text{f}}_{\text{i}}{\text{u}}_{\text{i}}$ 2-6 2 4 -8 -2 -4 6-10 3 8 -4 -1 -3 10-14 4 12 =A 0 0 0 14-18 3 16 4 1 3

Mean

Median:

Median of ungrouped data:

When data is in ascending order,

If n = total number of observations

Median

Median of grouped data:

Median =

lower limit of median class

total no. of observations

cumulative frequency of class previous to median class

frequency of median class

class size

For eg.

 Class Frequency $\left({\text{f}}_{\text{i}}\right)$ Cumulative frequency 2-6 2 2 6-10 3 5…..cf 10-14….median class 4……f 9 14-18 3 12

median class

lower limit of median class

cumulative frequency of class previous to median class

frequency of median class

class size

Median

Mode:

Mode of ungrouped data:

It is most frequently occurred value in the data.

Mode of grouped data:

Mode

lower limit of modal class

size of class interval

frequency of modal class

frequency of class preceding to modal class

frequency of succeeding to modal class

For e.g.

 Class Frequencies $\left({\text{f}}_{\text{i}}\right)$ 2-6 2 6-10 3 = ${\text{f}}_{\text{0}}$ 10-14….modal class 4 = ${\text{f}}_{\text{1}}$ 14-18 3 = ${\text{f}}_{\text{2}}$

Mode

lower limit of modal class

size of class interval

frequency of modal class

frequency of class preceding modal class

frequency of succeeding modal class

Mode

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:

 Class Frequency $\left({\text{f}}_{\text{i}}\right)$ Cumulative frequency of less than type 2-6 2 2 6-10 3 5 10-14 4 9 14-18 3 12

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

 Class Frequency $\left({\text{f}}_{\text{i}}\right)$ Cumulative frequency of more than type 2-6 2 12 6-10 3 10 10-14 4 7 14-18 3 3

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.