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:
$\begin{array}{|c|c|} \hline \text{Expenditure (in Rs)} & \text{Number of families} \\ \hline 1000-1500 & 24 \\ 1500-2000 & 40 \\ 2000-2500 & 33 \\ 2500-3000 & 28 \\ 3000-3500 & 30 \\ 3500-4000 & 22 \\ 4000-4500 & 16 \\ 4500-5000 & 7 \\ \hline \end{array}$
$\begin{array}{|c|c|} \hline \text{Expenditure (in Rs)} & \text{Number of families} \\ \hline 1000-1500 & 24 \\ 1500-2000 & 40 \\ 2000-2500 & 33 \\ 2500-3000 & 28 \\ 3000-3500 & 30 \\ 3500-4000 & 22 \\ 4000-4500 & 16 \\ 4500-5000 & 7 \\ \hline \end{array}$
(N/A) It can be observed from the given data that the maximum class frequency is $40$,belonging to $1500-2000$ interval.
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 modal class $= 24$.
Frequency $(f_2)$ of class succeeding modal class $= 33$.
Class size $(h) = 500$.
Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$
$= 1500 + \left( \frac{40 - 24}{2(40) - 24 - 33} \right) \times 500$
$= 1500 + \left( \frac{16}{80 - 57} \right) \times 500 = 1500 + \frac{8000}{23} \approx 1500 + 347.83 = 1847.83$.
Thus,the modal monthly expenditure is Rs. $1847.83$.
To find the mean,we use the step-deviation method:
Mean $(\bar{x}) = a + h \times \left( \frac{\sum f_i u_i}{\sum f_i} \right)$
Using $a = 2750$ and $h = 500$,we calculate $\sum f_i u_i = -35$ and $\sum f_i = 200$.
Mean $= 2750 + 500 \times \left( \frac{-35}{200} \right) = 2750 - 87.5 = 2662.5$.
Thus,the mean monthly expenditure is Rs. $2662.5$.
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 modal class $= 24$.
Frequency $(f_2)$ of class succeeding modal class $= 33$.
Class size $(h) = 500$.
Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$
$= 1500 + \left( \frac{40 - 24}{2(40) - 24 - 33} \right) \times 500$
$= 1500 + \left( \frac{16}{80 - 57} \right) \times 500 = 1500 + \frac{8000}{23} \approx 1500 + 347.83 = 1847.83$.
Thus,the modal monthly expenditure is Rs. $1847.83$.
To find the mean,we use the step-deviation method:
Mean $(\bar{x}) = a + h \times \left( \frac{\sum f_i u_i}{\sum f_i} \right)$
Using $a = 2750$ and $h = 500$,we calculate $\sum f_i u_i = -35$ and $\sum f_i = 200$.
Mean $= 2750 + 500 \times \left( \frac{-35}{200} \right) = 2750 - 87.5 = 2662.5$.
Thus,the mean monthly expenditure is Rs. $2662.5$.
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