Question

Explain all three formulas of Mean of the Grouped Data.

According to class 10.
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Answer 1

$\underline{\textsf {Arithmetic Mean-Grouped Frequency Distribution}}$

When the data are grouped in class intervals and presented in the form of a frequency table, we get a frequency distribution. In grouped frequency distribution, arithmetic mean may be computed by applying any one the following methods.

1. Direct Method
2. Assumed Mean Method
3. Step Deviation Method

Direct Method :

When direct method is used, the formula for finding the arithmetic mean is

$\boxed{\overline{X}=\dfrac{\Sigma fx}{\Sigma f}}$

Where $\sf x$ is the mid-point of the class interval and $\sf f$ is the corresponding frequency

Steps :

• Obtain the mid-point of each class and denote it by $\sf x$
• Multiply those mid-points by the respective frequency of each class and obtain the sum of $\sf fx$
• Divide $\sf \Sigma fx$ by $\sf \Sigma f$ to obtain mean

Assumed Mean Method :

if the observations are large, finding the products of the observations and their corresponding frequencies, and then adding them is not only difficult and time consuming but also has chances of errors. In such cases, we can use the Assumed Mean Method to find the arithmetic mean of grouped data. The formula for finding Arithmetic mean as by Assumed mean Method is :

$\boxed{ \overline{X}=A+\dfrac{\Sigma fd}{\Sigma f}}$

Steps :

1. Assume any value of the observations as the Mean $\sf (A)$. Preferably, choose the middle value.
2. Calculate the deviation $\sf d=x-A$ for each class
3. Multiply each of the corresponding frequency $\sf f$ with $\sf d$and obtain $\sf \Sigma fd$

Step Deviation Method :

In order to simplify the calculation, we divide the deviation by the width of class intervals $\sf \Big(i . e \:calculate \: \dfrac{x-A}{c} \Big)$and then multiply by c in the formula for getting the mean of the data. The formula to calculate the Arithmetic Mean is

$\sf \boxed{ \overline{X} = A + \Big[\dfrac{\Sigma fd}{\Sigma f}\times c\Big], \textsf{where} \Big(d = \dfrac{x-A}{c}\Big)}$