1. **Stating the problem:** We are given a frequency distribution table with class intervals and their corresponding frequencies. The problem is to understand and analyze this frequency distribution. 2. **Frequency distribution table:** | Interval | Frequency | |----------|-----------| | 14 - 22 | 8 | | 22 - 26 | 4 | | 26 - 30 | 4 | | 30 - 35 | 3 | | 35 - 39 | 1 | 3. **Explanation:** This table shows how many data points fall within each interval. For example, 8 data points fall between 14 and 22. 4. **Total frequency:** To find the total number of data points, sum all frequencies: $$ 8 + 4 + 4 + 3 + 1 = 20 $$ 5. **Midpoint of each class interval:** The midpoint $m$ of an interval $[a,b]$ is calculated as: $$ m = \frac{a + b}{2} $$ Calculate midpoints: - For 14 - 22: $\frac{14 + 22}{2} = 18$ - For 22 - 26: $\frac{22 + 26}{2} = 24$ - For 26 - 30: $\frac{26 + 30}{2} = 28$ - For 30 - 35: $\frac{30 + 35}{2} = 32.5$ - For 35 - 39: $\frac{35 + 39}{2} = 37$ 6. **Calculate the mean:** The mean $\bar{x}$ of grouped data is: $$ \bar{x} = \frac{\sum f_i m_i}{\sum f_i} $$ Where $f_i$ is frequency and $m_i$ is midpoint. Calculate numerator: $$ (8 \times 18) + (4 \times 24) + (4 \times 28) + (3 \times 32.5) + (1 \times 37) = 144 + 96 + 112 + 97.5 + 37 = 486.5 $$ Calculate mean: $$ \bar{x} = \frac{486.5}{20} = 24.325 $$ 7. **Interpretation:** The average value of the data set is approximately 24.33. This completes the analysis of the frequency distribution table.