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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 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

The class 1500 – 2000 has the maximum frequency. Therefore, it is the modal class.

Here l = 1500,
h = 500,
f1 =40 ,
f0 =24 and
f2 =33

Now, let us substitute these values in the formula

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

$$= \ 1500 \ + \ ( \frac{40 \ - \ 24}{2 \ × \ 40 \ - \ 24 \ - \ 33}) \ × \ 500$$

$$= \ 1500 \ + \ \frac{40 \ -\ 24}{80 \ - \ 24 \ - \ 33} \ × \ 500$$

$$= \ 1500 \ + \ \frac{16}{23} \ × \ 500$$

$$= \ 1500 \ + \ 347.83 \ = \ 1847.83$$

$$\therefore$$ Modal monthly expenditure = Rs. 1847.83

Let the assumed mean be A = 2750 and h = 500.

 Class Interval $$f_i$$ $$x_i$$ $$d_i \ = \ x_i \ - \ A$$ $$u_i \ = \ \frac{d_i}{h}$$ $$f_i u_i$$ 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 $$\sum f_i \ = \ 200$$ $$\sum f_i u_i \ = \ -35$$

The formula to calculate the mean,

$$\overline{x} \ = \ A \ + h \frac{ \sum \ f_i x_i}{ \sum f_i}$$

$$= \ 2750 \ + \ ( \frac{-35}{200}) \ × \ 500$$

$$= \ 2750 \ - \ 87.50 \ = \ 2662.50$$

So, the mean monthly expenditure of the families is Rs 2662.50