Question

The following table gives the distribution of total household expenditure (in rupees) of manual workers in the city

Expenditure	Frequency
100-150	24
150-200	40
200-250	33
250-300	28
300-350	30
350-400	22
400-450	16
450-500	7

Find the average expenditure (in Rs) per household.

Answer 1

Hint:
We are given the frequency of the expenditure and we need to find the average of the set of observations which is also called as the mean and the formula is given by
Mean $ = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$

Complete step by step solution:
So here we are given the distribution or the table which shows that or the expenditures and their frequency are distributed.

Expenditure	Frequency
100-150	24
150-200	40
200-250	33
250-300	28
300-350	30
350-400	22
400-450	16
450-500	7

So here we are given that between the expenditure of $100{\text{ and 150}}$ we have $24$ workers and similarly we are given all other ranges of the expenditures and number of persons lying in that range.
So here we saw how the expenditures of the total household is distributed among the manual workers in a city and we have to find the average expenditure of the workers. Average value is also termed as the mean value and its formula is given by
Mean$ = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$
Here ${x_1} = \dfrac{{100 + 150}}{2} = 125$
Similarly all other values will be the average of their upper and lower limit whenever we are given the terms in the form of the interval.
$
  {x_2} = \dfrac{{150 + 200}}{2} = 175 \\
  {x_3} = 225 \\
  {x_4} = 275 \\
  {x_5} = 325 \\
  {x_6} = 375 \\
  {x_7} = 425 \\
  {x_8} = 475 \\
 $
Similarly we are given the frequencies values as
$
  {f_1} = 24 \\
  {f_2} = 40 \\
  {f_3} = 33 \\
  {f_4} = 28 \\
  {f_5} = 30 \\
  {f_6} = 22 \\
  {f_7} = 16 \\
  {f_8} = 7 \\
 $
Now we apply the formula as
Mean$ = \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{f_i}} }}$
$ = \dfrac{{(24)(125) + 40(175) + 33(225) + 28(275) + 30(325) + 22(375) + 16(425) + 7(475)}}{{24 + 40 + 33 + 28 + 30 + 22 + 16 + 7}}$
$ = \dfrac{{53250}}{{200}} = 266.25$

Hence average value is $Rs266.25$

Note:
Mean or average can also be calculated by using the step deviation method in which we again have the formula of the mean as
Mean $ = a + \left( {\dfrac{{\sum {{f_i}{x_i}} }}{n}} \right)h$
Here $a, h, n$ represents the assumed mean, class size and the sum of the frequency.