Understanding the Problem
To find the mode of the given data, we first need to identify the class interval (CI) with the highest frequency (number of students). The data provided is:
- CI: 10 to 20, Number of Students: 238
- CI: 20 to 30, Number of Students: 814
- CI: 30 to 40, Number of Students: 832
- CI: 40 to 50, Number of Students: 220
- CI: 50 to 60, Number of Students: 110
- CI: 60 to 70, Number of Students: 70
- CI: 70 to 80, Number of Students: 20
Identifying the Modal Class
The class with the highest frequency is the modal class.
- The frequency distribution shows:
- 10 to 20: 238
- 20 to 30: 814
- 30 to 40: 832
- 40 to 50: 220
- 50 to 60: 110
- 60 to 70: 70
- 70 to 80: 20
- The maximum frequency is 832, which corresponds to the class interval 30 to 40.
Calculating the Mode
To calculate the mode using the formula for grouped data:
- Formula:
\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)} \right) \times h \]
Where:
- \( L \) = lower boundary of the modal class = 30
- \( f_1 \) = frequency of the modal class = 832
- \( f_0 \) = frequency of the class before the modal class = 814
- \( f_2 \) = frequency of the class after the modal class = 220
- \( h \) = class width = 10
Substituting the values:
- \[ \text{Mode} = 30 + \left( \frac{832 - 814}{(832 - 814) + (832 - 220)} \right) \times 10 \]
- \[ = 30 + \left( \frac{18}{18 + 612} \right) \times 10 \]
- \[ = 30 + \left( \frac{18}{630} \right) \times 10 \]
- \[ = 30 + 0.2857 \approx 30.29 \]
Conclusion
Thus, the mode of the given data is approximately 30.29. This value indicates the most frequently occurring range of student numbers in the specified intervals.