Question

The distribution of income of $60$teachers in a school is given below. Find the arithmetic mean of the following data using the shortcut method.

Daily wages	$150 - 200$	$200 - 250$	$250 - 300$	$300 - 350$	$350 - 400$
Number of teachers	$14$	$16$	$10$	$8$	$12$

Answer 1

The Arithmetic mean is simply the mean of given data calculated as: sum of observations divided by the number of observations. In shortcut method we assume mean by our self and use it in the formula \mathop X\limits^{\\_\\_} = A + \dfrac{{\sum {fd} }}{{\sum f }}.

Complete step by step solution:
The formula for finding arithmetic mean using shortcut method is \mathop X\limits^{\\_\\_} = A + \dfrac{{\sum {fd} }}{{\sum f }} where, \mathop X\limits^{\\_\\_} is the arithmetic mean, $A$ is assumed mean, $\sum f d$ and $\sum f$ are the summations.
As we know that the midpoint of $X$ is given by the formula $\dfrac{{upper\,range + lower\,range}}{2}$.
Now, use the above formula to find the mid values as,
The value of ${X_1}$ can be calculated as:
${X_1} = \dfrac{{150 + 200}}{2}$
$= \dfrac{{350}}{2}$
$= 175$
The value of ${X_2}$ can be calculated as:
${X_2} = \dfrac{{200 + 250}}{2}$
$= \dfrac{{450}}{2}$
$= 225$
The value of ${X_3}$ can be calculated as:
${X_3} = \dfrac{{250 + 300}}{2}$
$= \dfrac{{550}}{2}$
$= 275$
The value of ${X_4}$ can be calculated as:
${X_4} = \dfrac{{300 + 350}}{2}$
$= \dfrac{{650}}{2}$
$= 325$
The value of ${X_5}$ can be calculated as:
${X_5} = \dfrac{{350 + 400}}{2}$
$= \dfrac{{750}}{2}$
$= 375$

Now, assumed means $A = 275$. We can also assume any of the value from $X$. As per this assumption, mean will vary.

Now, refer the table given below for further calculation:

For $\sum {fd}$ multiply $f$ and $d$ for each column and add all the values in the column for $\sum f d$. The sum of this may be both positive and negative.
For $\sum f$ add all the values in the column $\sum f$. The sum of this will always be positive.

Daily wages	Midpoint of $X$	Number of teachers $f$	A = 275 \\\ d = x - A \\\	$fd$
$150 - 200$	$175$	$14$	$- 100$	$- 1400$
$200 - 250$	$225$	$16$	$- 50$	$- 800$
$250 - 300$	$275 = A$	$10$	$0$	$0$
$300 - 350$	$325$	$8$	$50$	$400$
$350 - 400$	$375$	$12$	$100$	$1200$
	$\sum f = 60$		$\sum f d = - 600$

Now we will substitute the values $\sum f d = - 600$ and $\sum f = 60$ in the formula \mathop X\limits^{\\_\\_} = A + \dfrac{{\sum {fd} }}{{\sum f }}
\mathop X\limits^\\_ = 275 + \dfrac{{ - 600}}{{60}}
Dividing $- 600$by $60$.
$= 275 + \left( { - 10} \right)$
$= 275 - 10$
$= 265$

Therefore, the arithmetic mean is $265$.
So, option (C) is the correct answer.

Note:
The mean can be calculated for both grouped and ungrouped data. Here we have calculated the mean of grouped data using the shortcut method. To find the mean of the data, other two methods can be used. One method is the direct method and second is Step-deviation method.