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# The following table 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.)Frequency1000-1500241500-2000402000-2500332500-3000283000-3500303500-4000224000-4500164500-50007

Last updated date: 11th Aug 2024
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Hint: First, take the mid values of each class as ${x_i}$ and frequency ${f_i}$. The mean value is equivalent to the fraction between the addition of a product of mid-value with frequency and the total frequency and the formula to calculate mode is $l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h$.

Complete Step by Step Solution:
We are given the monthly expenditure of families.
Let us assume that ${f_i}$ represents the number of consumers and ${x_i}$ is the difference in the intervals divided by 2.
Let the missing frequency be $x$.
The frequency distribution table for the given data is as follows:
 Class Frequency (${f_i}$) Mid-value (${x_i}$) ${f_i}{x_i}$ 1000-1500 24 1250 30000 1500-2000 40 1750 70000 2000-2500 33 2250 74250 2500-3000 28 2750 77000 3000-3500 30 3250 97500 3500-4000 22 3750 82500 4000-4500 16 4250 68000 4500-5000 7 4750 33250 Total $\sum {{f_i}} = 200$ $\sum {{f_i}{x_i}} = 532500$

We know that the general formula to find the mean value is,
Mean $= \dfrac{{\sum {{x_i}{f_i}} }}{{\sum {{x_i}} }}$
Now, we will substitute the value for the sum of the product of frequency and midpoint and the value for the sum of total frequency.
$\Rightarrow$ Mean $= \dfrac{{532500}}{{200}}$
Divide numerator by the denominator,
$\therefore$ Mean $= 2662.5$
Hence the mean expenditure is Rs. 2662.5.
The formula of mode is,
Mode $= l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h$
The modal class is the interval with the highest frequency.
$\Rightarrow$Modal class $= 1500 - 2000$
The lower limit of the modal class is,
$\Rightarrow l = 1500$
The class-interval is,
$\Rightarrow h = 1500 - 1000 = 500$
The frequency of the modal class is,
$\Rightarrow {f_1} = 40$
The frequency of the class before the modal class is,
$\Rightarrow {f_0} = 24$
The frequency of the class after modal class is,
$\Rightarrow {f_2} = 33$
Substitute these values in the mode formula,
$\Rightarrow$ Mode $= 1500 + \dfrac{{40 - 24}}{{2\left( {40} \right) - 24 - 33}} \times 500$
Simplify the terms,
$\Rightarrow$ Mode $= 1500 + \dfrac{{16}}{{80 - 57}} \times 500$
Subtract the values in the denominator and multiply the terms in the numerator,
$\Rightarrow$ Mode $= 1500 + \dfrac{{8000}}{{23}}$
Divide the numerator by denominator,
$\Rightarrow$ Mode $= 1500 + 347.83$
$\therefore$ Mode $= 1847.83$