Question 3
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.

$\begin{matrix} Expenditure (in Rs) & Number of families 1000 - 1500 & 24 1500 - 2000 & 40 2000 - 2500 & 33 2500 - 3000 & 28 3000 - 3500 & 30 3500 - 4000 & 22 4000 - 4500 & 16 4500 - 5000 & 7 \end{matrix}$

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Solution

From the given data, maximum class frequency is 40 belonging to 1500 - 2000 interval.

So, modal class = 1500 - 2000

Lower limit $l$ of modal class = 1500

Frequency $(f_{1})$ of modal class = 40

Frequency $(f_{0})$ of class preceding modal class = 24

Frequency $(f_{2})$ of class succeeding modal class = 33

Class size $h$ = 500

$M o d e = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h = 1500 + (\frac{40 - 24}{2 (40) - 24 - 33}) \times 500 = 1500 + (\frac{16}{80 - 57}) \times 500 = 1500 + \frac{8000}{23}$

= 1500 + 347.826

= 1847.826 = 1847.83

So, modal monthly expenditure was Rs.1847.83

Now, to calculate mean :
we may find class mark as;

$C l a s s m a r k = \frac{u p p e r c l a s s l i m i t + l o w e r c l a s s l i m i t}{2}$

Class size $h$ of given data = 500

Now, taking 2750 as assumed mean $a$ we may calculate $d_{i}, u_{i} a n d f_{i} u_{i}$ as following:

$\begin{matrix} E x p e n d i t u r e (i n R s) & N u m b e r o f f a m i l i e s f_{i} & x_{i} & d_{i} = x_{i} - 2750 & u_{i} = \frac{x_{i} - 2750}{h} & f_{i} u_{i} 1000 - 1500 & 24 & 1250 & - 1500 & - 3 & - 72 1500 - 2000 & 40 & 1750 & - 1000 & - 2 & - 80 2000 - 2500 & 33 & 2250 & - 500 & - 1 & - 33 2500 - 3000 & 28 & 2750 & 0 & 0 & 0 3000 - 3500 & 30 & 3250 & 500 & 1 & 30 3500 - 4000 & 22 & 3750 & 1000 & 2 & 44 4000 - 4500 & 16 & 4250 & 1500 & 3 & 48 4500 - 5000 & 7 & 4750 & 2000 & 4 & 28 T o t a l & 200 & - 35 \end{matrix}$

Now, from the table it may be observed that:

$\sum f_{i} = 200 \sum f_{i} u_{i} = - 35 ¯ x (m e a n) = a + (\frac{\sum f_{i} u_{i}}{\sum f_{i}}) \times h ¯ x = 2750 + (\frac{- 35}{200}) \times 500$

= 2750 - 87.5

= 2662.5

So, the mean monthly expenditure was Rs.2662.50.

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Similar questions

200

Expenditure (in Rs.)	Frequency	Expenditure (in Rs.)	Frequency
1000−1500 1500−2000 2000−2500 2500−3000	24 40 33 28	3000−3500 3500−4000 4000−4500 4500−5000	30 22 16 7

200

Monthly Expenditure	150-225	225-300	300-375	375-450	450-525	525-600	600 and above
No. of households	65	171	196	75	53	26	14

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