Question 4
The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures
| No of students per teacher | $\dfrac{Number of states} {U.T}$ |
|---|---|
| 15-20 | 3 |
| 20-25 | 8 |
| 25- 30 | 9 |
| 30-35 | 10 |
| 35-40 | 3 |
| 40-45 | 0 |
| 45- 50 | 0 |
| 50-55 | 2 |
Solution:
Given data:
Modal class = 30 – 35,
l = 30,
Class width (h) = 5,
$f_m$ = 10, $f_{1}$ = 9 and $f_{2}$ = 3
Mode Formula:| Mode = $l + [\dfrac{(f_m – f_{1})}{(2f_m – f_{1} – f_{2})}] × h$ |
Substitute the values in the given formula
Mode = 30 + $[\dfrac{(10 – 9)}{(20 – 9 – 3)}]$ × 5
= 30 + $(\dfrac{5}{8})$
= 30 + 0.625
= 30.625
Therefore, the mode of the given data = 30.625
Calculation of mean:
Find the midpoint using the formula, $x_{i} =\dfrac{(upper \: limit + lower \: limit)}{2}$
| Class Interval | Frequency $(f_{i})$ | Mid-point $(x_{i})$ | $f_{i}x_{i}$ |
|---|---|---|---|
| 15-20 | 3 | 17.5 | 52.5 |
| 20-25 | 8 | 22.5 | 180.0 |
| 25-30 | 9 | 27.5 | 247.5 |
| 30-35 | 10 | 32.5 | 325.0 |
| 35-40 | 3 | 37.5 | 112.5 |
| 40-45 | 0 | 42.5 | 0 |
| 45-50 | 0 | 47.5 | 0 |
| 50-55 | 2 | 52.5 | 105.0 |
| Sum $f_i$ = 35 | Sum $f_{i}x_{i}$=1022.5 |
| Mean = $\overline{x} = \dfrac{∑f_{i}x_{i}}{∑f_{i}}$ |
= $\dfrac{1022.5}{35}$
= 29.2 (approx)
Therefore, mean = 29.2