1. **Problem Statement:** We have a dataset of exam scores for students: $$67,78,87,90,50,66,78,70,80,80,80,91,92,93,96,90,97,77,78,79,87,89,87,89,80,83,85,88,89,75,76,90,77,88,88,76,76,83,83,84,79,80,69,90,98,79,90,89,89,79,90,90,85,85,86,87,81,81,81,81,78,87,90,89,76,66,78,78,91,88,81,81,82,83,85,76,75,75,79,78,76,69,69,80,79,89,80,80,86,86,87,88,89,91,90,97,96,98,80,80$$ We need to: - Tally the data into frequency intervals - Find Mode, Median, Mean - Calculate Standard Deviation 2. **Tally Frequency Table:** Intervals: 50-59, 60-69, 70-79, 80-89, 90-100 Count scores in each interval: - 50-59: Scores = 50 → count = 1 - 60-69: Scores = 66,66,69,69,69 → count = 5 - 70-79: Scores = 70,75,75,75,76,76,76,76,77,77,78,78,78,78,78,79,79,79,79 → count = 19 - 80-89: Scores = 80,80,80,80,80,80,80,80,81,81,81,81,81,82,83,83,83,84,85,85,85,86,86,86,87,87,87,87,88,88,88,88,89,89,89,89,89 → count = 37 - 90-100: Scores = 90,90,90,90,90,90,90,90,91,91,91,92,93,96,96,97,97,98,98 → count = 19 3. **Mode:** Mode is the most frequent score(s). From tally, 80 appears 8 times, 89 appears 5 times, 90 appears 8 times. Mode = 80 and 90 (bimodal) 4. **Median:** Sort data and find middle value. Total data points = 100. Median = average of 50th and 51st values. Sorted data median values are both 85. Median = 85 5. **Mean:** Sum all scores and divide by 100. Sum = 8280 Mean = \frac{8280}{100} = 82.8 6. **Standard Deviation:** Formula: $$\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2}$$ Calculate squared differences, sum = 1056.4 $$\sigma = \sqrt{\frac{1056.4}{100}} = \sqrt{10.564} \approx 3.25$$ **Final answers:** - Frequency tally: 50-59:1, 60-69:5, 70-79:19, 80-89:37, 90-100:19 - Mode: 80 and 90 - Median: 85 - Mean: 82.8 - Standard Deviation: 3.25