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 C)	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

Hint: According to the question, we need to find out the mode and mean. For mode, we need to check that if we have all the values according to its formula. The highest frequency, etc, . Same for mean also. After finding the values, we have to simply put the values according to the formula and solve it.

Formula used:
\[Mode = l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h\] and \[Mean = a + h \times \dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}\]

Complete step by step solution:
From the given table, we get that the highest frequency is 40. This frequency is represented as ‘$f_1$’. The preceding frequency which is represented as ‘$f_0$’ is 24. The succeeding frequency which is represented as ‘$f_2$’ is 33.
The interval for having the highest frequency is \[1500 - 2000\] . We call this interval a Modal class. So, in this class the lower limit which is represented as ‘l’ is 1500. Now, the class size is represented as ‘h’. We calculate it by taking the difference between the intervals. So:
 \[h = 2000 - 1500\]
 \[ \Rightarrow h = 500\]
Now, we will calculate the mode. The formula for calculating mode is:
 \[Mode = l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h\]
Now, we will put the values according to the formula, and we get:
 \[ \Rightarrow Mode = 1500 + \dfrac{{40 - 24}}{{2 \times 40 - 24 - 33}} \times 500\]
 \[ \Rightarrow Mode = 1500 + \dfrac{{16}}{{80 - 57}} \times 500\]
 \[ \Rightarrow Mode = 1500 + \dfrac{{16}}{{23}} \times 500\]
 \[ \Rightarrow Mode = 1500 + 347.826\]
 \[ \Rightarrow Mode = 1847.83\]
Therefore, the modal monthly expenditure of the families is \[1847.83\] .
Now, we will calculate the mean. The formula for calculating the mean is:
 \[ \Rightarrow Mean = a + h \times \dfrac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}\]
Here, ‘a’ is the assumed mean whose value is 3250, ‘h’ is the class interval and the value is 500, \[{u_i} = \dfrac{{{x_i} - a}}{h}\] , and \[{f_i} = 200\] . When we calculate \[\sum {{f_i}{u_i}} \] , we get the value as \[ - 235\] .
Now, we will put these values in the formula, and we get:
 \[ \Rightarrow Mean = 3250 + 500 \times \dfrac{{ - 235}}{{200}}\]
 \[ \Rightarrow Mean = 3250 + 5 \times \dfrac{{ - 235}}{2}\]
 \[ \Rightarrow Mean = 3250 - 587.2\]
 \[ \Rightarrow Mean = 2662.5\]
Therefore, we got our mean as \[2662.5\] .
So, the correct answer is “ \[2662.5\] ”.

Note: In Mathematics, this mean and mode topics comes under the chapter called Statistics. Statistics is a study where we have a collection, presentation, organization, analysis of data. There are many more formulas used other than mean and mode, they are variance, standard deviation, range, etc, .