Question

The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure :

Expenditure (in Rs)	Number of families
1000 - 1500	24
1500 - 2000	40
2000 - 2500	33
2500 - 3000	28
3000 - 3500	30
3500 - 4000	22
4000 - 4500	16
4500 - 5000	7

Answer 1

It can be observed from the given data that the maximum class frequency is 40, belonging to 1500 − 2000 intervals. 

Therefore, modal class =  1500 - 2000
Lower limit (\(l\)) of modal class = 1500
Frequency (\(f_1\)) of modal class = 40
Frequency (\(f_0\)) of class preceding the modal class = 24
Frequency (\(f_2\)) of class succeeding the modal class = 33
Class size (\(h\)) = 500

Mode = \(l\) + \((\frac{f_1 - f_0 }{2f_1 - f_0 - f_2)} \times h\)

Mode =  \(1500 + (\frac{40 - 24 }{ 2(40) - 24 - 33}) \times 500\)

Mode =\(1500+ [\frac{16}{80 - 57}] \times 500\)

Mode = \(1500 +( \frac{8000}{ 23})\)
Mode = 1500 + 347.826
Mode = 1847.826
Mode = 1847.83

Therefore, modal monthly expenditure was Rs 1847.83.  

---

To find the class mark (\(x_i\)) for each interval, the following relation is used.  

Class mark  \((x_i)\) = \(\frac {\text{Upper \,limit + Lower \,limit}}{2}\)

class size (h) of the data = 500

Taking 2750 as assured mean (a), \(d_i\), \(u_i\), and \(f_iu_i\) can be calculated as follows.   

Expenditure (in  Rs)	Number of families  (fi)	\(\bf{x_i}\)	\(\bf{d_i = x_i -2750}\)	\(\bf{u_i = \frac{d_i}{500}}\)	\(\bf{f_iu_i}\)
1000 - 1500	24	1250	-1500	-3	-72
1500 - 2000	40	1750	-1000	-2	-80
2000 - 2500	33	2250	-500	-1	-33
2500 - 3000	28	2750	0	0	0
3000 - 3500	30	3250	500	1	30
3500 - 4000	22	3750	1000	2	44
4000 - 4500	16	4250	1500	3	48
4500 - 5000	7	4750	2000	4	28
Total	200				-35

From the table, it can be observed that  

\(\sum f_i = 200\)
\(\sum f_iu_i = -35\)

Mean, \(\overset{-}{x} = a + (\frac{\sum f_iu_i}{\sum f_i})h\) 

x = \(2750 + (\frac{-352 }{200})\times 500\)

x = 2750 - 87.5
x = 2662.5

Therefore, mean monthly expenditure was Rs 2662.50.