Question 4

Let's find the mode and mean of the given data.

Step 1 — Find the Mode

First, we identify the modal class. The maximum frequency is 10. This frequency belongs to the class interval 30 - 35. So, the modal class is 30 - 35.

We list the values for the mode formula: Lower limit of modal class ($l$) is 30. Class size ($h$) is 5. Frequency of modal class ($f_1$) is 10. Frequency of class preceding modal class ($f_0$) is 9. Frequency of class succeeding modal class ($f_2$) is 3.

Now, we use the mode formula: $\text{Mode} = l + \left[ \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right] \times h$

Let's substitute the values into the formula. $\text{Mode} = 30 + \left[ \frac{10 - 9}{2 \times 10 - 9 - 3} \right] \times 5$

$= 30 + \left[ \frac{1}{20 - 12} \right] \times 5$

$= 30 + \left[ \frac{1}{8} \right] \times 5$

$= 30 + 0.125 \times 5$

$= 30 + 0.625$

$\boxed{\text{Mode} = 30.625}$

This means that most of the states or U.T. have a teacher-student ratio of approximately 30.6.

Step 2 — Find the Mean

Let's use the step deviation method to find the mean. We first find the class mark ($x_i$) for each interval. The class mark is the midpoint of the class interval. Let's choose an assumed mean ($a$) as 32.5. The class size ($h$) is 5.

We create a table to organize our calculations:

| Number of students per teacher | Number of states/U.T ($f_i$) | $x_i$ | $d_i = x_i - 32.5$ | $u_i = d_i / 5$ | $f_i u_i$ | | :----------------------------- | :---------------------------: | :---: | :----------------: | :-------------: | :-------: | | 15 - 20 | 3 | 17.5 | -15 | -3 | -9 | | 20 - 25 | 8 | 22.5 | -10 | -2 | -16 | | 25 - 30 | 9 | 27.5 | -5 | -1 | -9 | | 30 - 35 | 10 | 32.5 | 0 | 0 | 0 | | 35 - 40 | 3 | 37.5 | 5 | 1 | 3 | | 40 - 45 | 0 | 42.5 | 10 | 2 | 0 | | 45 - 50 | 0 | 47.5 | 15 | 3 | 0 | | 50 - 55 | 2 | 52.5 | 20 | 4 | 8 | | Total | 35 | | | | -23 |

From the table, we have: $\sum f_i = \textbf{35}$ $\sum f_i u_i = \textbf{-23}$ Assumed mean ($a$) = 32.5 Class size ($h$) = 5

Now, we apply the mean formula for the step deviation method: $\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$

Let's substitute the values. $\bar{x} = 32.5 + \left( \frac{-23}{35} \right) \times 5$

$= 32.5 - \frac{23 \times 5}{35}$

$= 32.5 - \frac{115}{35}$

$= 32.5 - \frac{23}{7}$

$= 32.5 - 3.2857...$

$\boxed{\text{Mean} \approx 29.22}$

This means that, on average, the teacher-student ratio across the states is approximately 29.2.

Answer

(i) The mode of the data is 30.625. (ii) The mean of the data is approximately 29.22. (iii) The mode indicates that the most frequent teacher-student ratio among the states/U.T. is approximately 30.6. The mean indicates that the average teacher-student ratio across all states/U.T. is approximately 29.2.