1. **Problem Statement:** We have a frequency distribution of scores in intervals and their corresponding frequencies. We want to understand the data and possibly find measures like the mean or visualize it. 2. **Data Given:** - Scores intervals: 46-50, 41-45, 36-40, 31-35, 26-30, 21-25 - Frequencies: 4, 6, 10, 8, 12, 5 respectively 3. **Step 1: Find the midpoints of each score interval.** The midpoint $m$ of an interval $[a,b]$ is given by: $$m = \frac{a+b}{2}$$ Calculate midpoints: - $\frac{46+50}{2} = 48$ - $\frac{41+45}{2} = 43$ - $\frac{36+40}{2} = 38$ - $\frac{31+35}{2} = 33$ - $\frac{26+30}{2} = 28$ - $\frac{21+25}{2} = 23$ 4. **Step 2: Use midpoints and frequencies to find the mean score.** The formula for mean of grouped data is: $$\bar{x} = \frac{\sum (f_i \cdot m_i)}{\sum f_i}$$ where $f_i$ is frequency and $m_i$ is midpoint. Calculate numerator: $$4 \times 48 + 6 \times 43 + 10 \times 38 + 8 \times 33 + 12 \times 28 + 5 \times 23$$ $$= 192 + 258 + 380 + 264 + 336 + 115 = 1545$$ Calculate denominator: $$4 + 6 + 10 + 8 + 12 + 5 = 45$$ Calculate mean: $$\bar{x} = \frac{1545}{45} = 34.33$$ 5. **Interpretation:** The average score is approximately 34.33. 6. **Additional:** The bar graph described shows frequencies on the y-axis and score intervals on the x-axis, matching the frequency table. This analysis helps understand the distribution and central tendency of the scores.