1. **Problem Statement:** Compute the mean, median, mode, range, and sample variance of the reading test scores: 72, 88, 81, 75, 88, 91, 84, 75, 90, 80. 2. **Mean:** The mean is the average of all scores. Formula: $$\text{Mean} = \frac{\sum x_i}{n}$$ where $x_i$ are the scores and $n$ is the number of scores. Calculate sum: $72 + 88 + 81 + 75 + 88 + 91 + 84 + 75 + 90 + 80 = 824$ Number of scores: $n = 10$ Mean: $$\frac{824}{10} = 82.4$$ 3. **Median:** The median is the middle value when scores are ordered. Order scores: $72, 75, 75, 80, 81, 84, 88, 88, 90, 91$ Since $n=10$ (even), median is average of 5th and 6th scores. 5th score: $81$, 6th score: $84$ Median: $$\frac{81 + 84}{2} = 82.5$$ 4. **Mode:** The mode is the most frequent score. Frequencies: 75 appears 2 times, 88 appears 2 times, others less. Modes: $75$ and $88$ (bimodal) 5. **Range:** The range is the difference between max and min scores. Max score: $91$, Min score: $72$ Range: $$91 - 72 = 19$$ 6. **Sample Variance:** Measures spread of scores. Formula: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ where $\bar{x}$ is the mean. Calculate squared deviations: $(72 - 82.4)^2 = 108.16$ $(75 - 82.4)^2 = 54.76$ (twice) $(80 - 82.4)^2 = 5.76$ $(81 - 82.4)^2 = 1.96$ $(84 - 82.4)^2 = 2.56$ $(88 - 82.4)^2 = 31.36$ (twice) $(90 - 82.4)^2 = 57.76$ $(91 - 82.4)^2 = 73.96$ Sum squared deviations: $108.16 + 54.76 + 54.76 + 5.76 + 1.96 + 2.56 + 31.36 + 31.36 + 57.76 + 73.96 = 422.4$ Divide by $n-1=9$: $$s^2 = \frac{422.4}{9} = 46.93$$ 7. **Interpretation:** - Mean and median are close ($82.4$ and $82.5$), indicating a symmetric distribution. - Mode shows two common scores, $75$ and $88$, suggesting some clustering. - Range of $19$ shows moderate spread. - Variance $46.93$ indicates moderate variability. Overall, the class performance is average with some variation but fairly consistent.