Mean of Grouped Data

In Statistics Mean, Median and Mode are known as the measures of central tendencies.
The mean of a given data is given by the sum of values of all the given observations divided by the number of observations.

Mean of a given data ($\stackrel{\_}{\text{x}}$)=  $\frac{\text{The sum of values of all given observations}}{\text{The number of observations}}$.

There are three methods to find the mean of a grouped data are:

• Direct Method
• Assumed Mean Method
• Step-deviation Method

Direct method:

Mid-point or Class mark (x_i) = $\frac{\text{[ Upper class limit + Lower class limit]}}{\text{2}}$

Mean of a given data using direct method ($\stackrel{\_}{\text{x}}$) = $\frac{\sum \text{f i x i}}{\sum \text{f i}}$   .

Where f_i is frequency of the given class interval.

Assumed Mean Method:

Mean of grouped data using assumed mean method ($\stackrel{\_}{\text{x}}$) = A + $\frac{\sum \text{f i d i}}{\sum \text{f i}}$

Where d_i = x_i - A.

A is an Assumed Mean.

f_i is frequency of the given class interval.

d_i is Deviation of the assumed mean from a class mark.

Step-deviation Method:

Mean of grouped data using step-deviation method ($\stackrel{\_}{\text{x}}$) = A + h [ $\frac{\sum \text{f i u i}}{\sum \text{f i}}$]

Where u_i = $\frac{{\text{x}}_{\text{i}}\text{- A}}{\text{h}}$

A is an Assumed Mean.

f_i is frequency of the given class interval.

h is class size.

• The step deviation method is more suitable, if the deviations of the class marks from the assumed mean are larger.
• In step-deviation method, the mean of grouped data remains the same for any non-zero numbers  and , such that u_i = $\frac{{\text{x}}_{\text{i}}\text{- A}}{\text{h}}$.