Question

Calculate the average daily income (in rupees) of the following data about men working in a company:

Daily Income (₹)	<100	<200	<300	<400	<500
Number of men	12	28	34	41	50

Answer 1

Here, we will calculate the mean that is average value by using the formula $\overline X = \left( {a + \dfrac{{{\sum _i}{f_i}{u_i}}}{{{\sum _i}{f_i}}} \times h} \right)$ and we will calculate mid value i.e $A = 250$, $\sum {f_i},{x_i},{u_i},{f_i}{u_i}$ and $\sum {f_i}{u_i}$
Formula Used: $\overline X = \left( {a + \dfrac{{{\sum _i}{f_i}{u_i}}}{{{\sum _i}{f_i}}} \times h} \right)$ and also calculate ${u_i} = \dfrac{{{x_i} - A}}{h}$
Here, $\overline X =$ means.
$A$ and $h$ are arbitrary constants and ${f_i}$ is the frequency.

Complete step by step solution:
From the given data, we can calculate $h$by using the above formula i.e $h = {x_n} - {x_{n - 1}}$ which comes out to be $h = 100 - 0 = 100$ So, $h = 100$ and here A is the mid value which is calculated from ${x_i}$ i.e calculated from the given intervals which is mid value of $0{\rm{ }}and{\rm{ }}100 = 50$. Similarly, calculated all the values of ${x_i}$.Now, next step is to calculate ${u_i}$from the above formula i.e ${u_i} = \dfrac{{{x_i} - A}}{h}$. Put h as 100 , A as 250 and ${x_i}$ has different values for different intervals. Then, calculate ${f_i}{u_i}$. At the end calculate the sum of ${f_i}$ , ${f_i}{u_i}$.

Class Interval (rupees)	Number of men $\left( {{f_i}} \right)$	${x_i}$	${u_i} = \dfrac{{{x_i} - A}}{h}$	${f_i}{u_i}$
$0 - 100$	$12$	$50$	$- 2$	$- 24$
$100 - 200$	$16$	$150$	$- 1$	$- 16$
$200 - 300$	$6$	$250 \to A$	$0$	$0$
$300 - 400$	$7$	$350$	$1$	$7$
$400 - 500$	$9$	$450$	$2$	$18$
$50$			$- 15$

As, we will use the formula of step deviation of mean that is $\overline X = \left( {a + \dfrac{{{\sum _i}{f_i}{u_i}}}{{{\sum _i}{f_i}}} \times h} \right)$ to calculate average daily income (in rupees)
After substituting all the calculated values we get,
$\overline X = \left( {250 + \dfrac{{\left( { - 15} \right)}}{{50}} \times 100} \right)$
On simplifying we get,
$\overline X = \left( {250 - 30} \right)$
$\overline X = 220$

Hence, the average daily income of the men is 220 rupees.

Note:
Note: To solve these types of questions you can use the alternative method by just calculating ${f_i}{x_i}$ and using the formula of mean that is $\overline X = \dfrac{{\sum {x_i}{f_i}}}{{\sum {f_i}}}$ as shown below.

Class Interval (rupees)	Number of men $\left( {{f_i}} \right)$	${x_i}$	${f_i}{x_i}$
$0 - 100$	$12$	$50$	$600$
$100 - 200$	$16$	$150$	$2400$
$200 - 300$	$6$	$250$	$1500$
$300 - 400$	$7$	$350$	$2450$
$400 - 500$	$9$	$450$	$4050$
$50$		$11,000$

Now, we will calculate Mean $\overline X = \dfrac{{\sum {x_i}{f_i}}}{{\sum {f_i}}}$ .
By substituting all the calculated values we get,
$\overline X = \dfrac{{11000}}{{50}}$
After dividing we get,
$\overline X = 220$
Hence, average income comes out to be 220 rupees.