Understanding the Data
The given data presents a frequency distribution of students across various class intervals (CIs). The intervals and their corresponding number of students are as follows:
- CI: 10 to 20 - Students: 2
- CI: 20 to 30 - Students: 3
- CI: 30 to 40 - Students: 8
- CI: 40 to 50 - Students: 14
- CI: 50 to 60 - Students: 8
- CI: 60 to 70 - Students: 3
- CI: 70 to 80 - Students: 2
Identifying the Mode
The mode is defined as the value that appears most frequently in a data set. In the context of grouped data like this, the mode can be found in the class interval with the highest frequency.
Step-by-Step Calculation
1. Find the Highest Frequency:
- The class interval 40 to 50 has the highest frequency of 14 students.
2. Determine the Modal Class:
- The modal class is 40 to 50.
3. Apply the Mode Formula for Grouped Data:
The formula to calculate the mode in grouped data is:
\[
\text{Mode} = L + \left( \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)} \right) \times h
\]
Where:
- \(L\) = lower boundary of the modal class (40)
- \(f_1\) = frequency of the modal class (14)
- \(f_0\) = frequency of the class before the modal class (8)
- \(f_2\) = frequency of the class after the modal class (8)
- \(h\) = width of the class intervals (10)
4. Substituting the Values:
\[
\text{Mode} = 40 + \left( \frac{14 - 8}{(14 - 8) + (14 - 8)} \right) \times 10
\]
Simplifying this gives:
\[
\text{Mode} = 40 + \left( \frac{6}{12} \right) \times 10 = 40 + 5 = 45
\]
Conclusion
The mode of the given data is 45, indicating that the most common value in the data set falls within the interval of 40 to 50.