माध्य - प्रत्यक्ष विधि

To calculate the mean of grouped data we have three different methods - direct method, assumed mean method, and step deviation method. The mean of grouped data deals with the frequencies of different observations or variables that are grouped together.

Direct Method

The direct method is the simplest method to find the mean of the grouped data. If the values of the observations are$x_{1},x_{2},x_{3}.............x_{n}$ with their corresponding frequencies are $f_{1},f_{2},f_{3}.............f_{n}$ then the mean of the data is given by,

${\bar {x}}={\frac {x_{1}f_{1}+x_{2}f_{2}+x_{3}f_{3}+......+x_{n}f_{n}}{f_{1}+f_{2}+f_{3}+......+f_{n}}}$

${\bar {x}}={\frac {\sum _{i=1}^{n}\displaystyle x_{i}f_{i}}{\sum _{i=1}^{n}\displaystyle f_{i}}}$

Here are the steps to find the mean for grouped data using the direct method,

• Create a table containing four columns such as class interval, class marks (corresponding), denoted by $x_{i}$, frequencies $f_{i}$ (corresponding), and $x_{i}f_{i}$.
• Calculate Mean by the Formula Mean = ${\bar {x}}={\frac {\sum _{i=1}^{n}\displaystyle x_{i}f_{i}}{\sum _{i=1}^{n}\displaystyle f_{i}}}$. Where $f_{i}$ is the frequency and $x_{i}$ is the midpoint of the class interval.
• Calculate the midpoint, $x_{i}$ using the formula $x_{i}$ = (upper class limit + lower class limit) / 2.

Example: Find the mean of the following data.

Class Interval	Frequency $f_{i}$
0 - 10	9
10 - 20	13
20 - 30	8
30 - 40	15
40 - 50	10

Step 1:

Class Interval	Frequency $f_{i}$	Class Mark $x_{i}$	$x_{i}$$f_{i}$
0 - 10	9	5	45
10 - 20	13	15	195
20 - 30	8	25	200
30 - 40	15	35	525
40 - 50	10	45	450
Total	55		1415