1. **Problem Statement:** We have exam scores of students and need to compute descriptive statistics: frequency tally, mode, median, mean, and standard deviation. 2. **Data:** 67,78,87,90,50,66,78,70,80,80,80,91,92,93,96,90,97,77,78,79,87,89,87,89,80,83,85,88,89,75,76,90,77,88,88,76,76,83,83,84,79,80,69,90,98,79,90,89,89,79,90,90,85,85,86,87,81,81,81,81,78,87,90,89,76,66,78,78,91,88,81,81,82,83,85,76,75,75,79,78,76,69,69,80,79,89,80,80,86,86,87,88,89,91,90,97,96,98,80,80 3. **Step 1: Frequency Tally by Score Ranges** Ranges: 50-59, 60-69, 70-79, 80-89, 90-100 Count scores in each range: - 50-59: 1 (50) - 60-69: 5 (66,66,67,69,69,69) - 70-79: 17 (70,75,75,75,76,76,76,76,76,77,77,78,78,78,78,78,79,79,79,79) - 80-89: 38 (80,80,80,80,80,80,80,81,81,81,81,81,82,83,83,83,84,85,85,85,86,86,86,87,87,87,87,88,88,88,88,89,89,89,89,89) - 90-100: 24 (90,90,90,90,90,90,90,90,90,90,91,91,91,92,93,96,96,97,97,98,98) 4. **Step 2: Calculate Mode** Mode is the most frequent score. Count frequencies: - 80 appears 8 times - 90 appears 10 times - 89 appears 6 times - 87 appears 4 times - 81 appears 5 times Mode = 90 5. **Step 3: Calculate Median** Sort data (already given sorted for median calculation): Number of data points $n=85$ Median position = $\frac{n+1}{2} = \frac{85+1}{2} = 43^{rd}$ value Sorted data 43rd value is 80 Median = 80 6. **Step 4: Calculate Mean** Mean formula: $$\text{Mean} = \frac{\sum x_i}{n}$$ Sum all scores: $$\sum x_i = 6773$$ Mean: $$\frac{6773}{85} \approx 79.68$$ 7. **Step 5: Calculate Standard Deviation** Formula: $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$ Calculate squared deviations and sum: $$\sum (x_i - 79.68)^2 = 3936.96$$ Standard deviation: $$s = \sqrt{\frac{3936.96}{84}} = \sqrt{46.82} \approx 6.84$$ **Final answers:** - Frequency tally: 50-59:1, 60-69:5, 70-79:17, 80-89:38, 90-100:24 - Mode: 90 - Median: 80 - Mean: 79.68 - Standard Deviation: 6.84