Question

Calculate the average income of workers using direct method:

Workers	4	6	8	10	12
Weekly Income	300	500	700	800	900

A) 10 rupees
B) 9 rupees
C) 8 rupees
D) 7 rupees

Answer 1

Hint: Average means mean. So, in this question we have to find the average income which means we have to find the mean income of workers. Since the method to be used to find the average income is already, mentioned, that is, we have to use the Direct Method for calculation of average or mean income, which is given as: $mean = \dfrac{{sum\,of\,all\,observations}}{{total\,number\,of\,observations}}$. For a frequency table, this formula gets transformed into $\overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$.

Complete step by step answer:
Now, according to the question we are required to find the Mean income of workers using the Direct Method.
The direct method for Mean is given as:
$\overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$
Where $\overline x $ is the mean and $\sum {{f_i}{x_i}} $ is the total sum of the frequency ${f_i}$ of weekly income multiplied with ${x_i}$ of workers for the given data.
Now, from the information in the question, we need to construct a table to apply the Direct formula in order to find the average or mean income. So, the table will be:

Workers(${x_i}$)	Weekly Income (${f_i}$)	${f_i}{x_i}$
4	300	1200
6	500	3000
8	700	5600
10	800	8000
12	900	10800
	$\sum {{f_i}} $=3200	$\sum {{f_i}{x_i}} $=28600

Clearly, from the table we have the values for $\sum {{f_i}} $is ₹3200, and $\sum {{f_i}{x_i}} $is ₹28600, so putting these values in the formula for Mean$(\overline x )$ by Direct Method, we will get the Mean or Average income of workers as:
$
  \overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }} \\
   = \dfrac{{28600}}{{3200}} \\
   = 8.93 \approx 9 \\
$
That is, 9 rupees.
So, for the given data the average income of workers is 9 rupees.

Therefore the correct answer is option B.

Note: There are other methods also to calculate the Mean for a given frequency table apart from the Direct Method. The other methods are – Assumed Mean Method and Step-deviation method. The formula for Assumed mean method is given as: $\overline x = a + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}$ where a is the assumed mean and ‘d’ is the deviation from the assumed mean of the observations given as: ${d_i} = {x_i} - a$ . The formula for Step-deviation method is given as: $\overline x = a + \dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }} \times h$, where ${u_i} = \dfrac{{{x_i} - a}}{h}$ , the deviation over the class.