Mean of Grouped Data
Mean of given set of data is given by the sum of values of all the given observations divided by the number of observations.
The three ways of finding the mean of grouped data are:
* Direct Method
* Assumed Mean Method
* Step-Deviation Method
Class Mark ( X_{i} ) = \frac{( upper \thinspace \thinspace Class \thinspace \thinspace Limit + Lower \thinspace \thinspace Class \thinspace \thinspace Limit)}{2}
Using direct method, Mean x̅ = \frac{ \sum f_{i} x_{i} }{ \sum f_{i} }
when x_{i} ‘s are mid values
Using assumed mean method,x̅ = a + \frac{ \sum f_{i} d_{i} }{ \sum f_{i} }
where a = assumed mean and d_{i} = x_{i} – a
Using step deviation method, x̅ = a + h × \frac{ \sum f_{i} u_{i} }{ \sum f_{i} }
where a = assumed mean, h = class size and u_{i} = \frac{ d_{i} }{ h }
Mode of Grouped Data
Mode of grouped data = ɭ + h × \frac{ ( f_{1} - f_{0} ) }{ ( 2f_{1} - f_{0} - f_{2} )}
Where:
* ɭ → lower class limit of the modal class (class interval with maximum frequency)
* H → class size (assuming all class size are equal)
* f_{1}→ frequency of the modal class.
* f_{0}→ frequency of class preceding the modal class.
* f_{2}→frequency of the class succeeding the modal class
Median of Grouped Data
Median of grouped data = ɭ + [ \frac{ (\frac{n}{2}) - ncf}{f} ] × h,
where:
* ɭ = lower class limit of median class
(class interval with cumulative frequency greater than and closest to \frac{n}{2} )
* n = total number of observations (sum of frequencies of all class intervals)
* cf = cumulative frequency of class preceding the median class
* f = frequency of the median class
* h = class size (assuming all class sizes are equal)
Graphical Representation of Cumulative Frequency Distribution
* A curve showing the cumulative frequency distribution is called an ogive.
* An ogive representing a cumulative frequency distribution of the more than type is called a ‘more than ‘ogive.
* An ogive representing a cumulative frequency distribution of less than type is called a ‘less than’ ogive.
* Ogive can be used to find the median grouped data.