Question 3

Let's find the modal and mean monthly expenditure for the families.

Step 1 — Find the Mode

First, we find the modal class. The modal class has the highest frequency. The highest frequency is 40. This belongs to the class 1500 - 2000. So, the modal class is 1500 - 2000.

We identify the values for the mode formula. The lower limit ($l$) of the modal class is 1500. The frequency ($f_1$) of the modal class is 40. The frequency ($f_0$) of the class before the modal class is 24. The frequency ($f_2$) of the class after the modal class is 33. The class size ($h$) is $2000 - 1500 = \textbf{500}$.

We use the formula for the mode of grouped data. Mode $= l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$

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

$= 1500 + \left( \frac{16}{80 - 24 - 33} \right) \times 500$

$= 1500 + \left( \frac{16}{23} \right) \times 500$

$= 1500 + \frac{8000}{23}$

$= 1500 + 347.8260...$

$\boxed{₹1847.83}$

Step 2 — Find the Mean

Let's use the step-deviation method to find the mean. We need to calculate class marks ($x_i$). We choose an assumed mean ($a$) from the class marks. Let's pick 2750. The class size ($h$) is 500.

Here is the table for calculations:

| Expenditure (in ₹) | Number of families ($f_i$) | Class Mark ($x_i$) | $d_i = x_i - 2750$ | $u_i = d_i / 500$ | $f_i u_i$ | | :----------------- | :------------------------- | :----------------- | :----------------- | :---------------- | :-------- | | 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 | | Total | $\sum f_i = 200$ | | | | $\sum f_i u_i = -35$ |

We use the formula for the mean by step-deviation method. Mean ($\bar{x}$) $= a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$

$= 2750 + \left( \frac{-35}{200} \right) \times 500$

$= 2750 + (-0.175) \times 500$

$= 2750 - 87.5$

$\boxed{₹2662.50}$

Answer

(i) The modal monthly expenditure is ₹1847.83. (ii) The mean monthly expenditure is ₹2662.50.