1. **Problem Statement:** Given the scores of the top 20 students in a 50-item statistics test: 27, 30, 45, 47, 50, 38, 29, 42, 31, 46, 48, 43, 38, 49, 38, 35, 37, 32, 44, 40. We need to: a. Construct a frequency distribution table. b. Determine the sample mean, median (Md), and mode (Mo). c. Calculate the variance and standard deviation. 2. **Step a: Frequency Distribution Table** - List unique scores and count their occurrences. | Score | Frequency | |-------|-----------| | 27 | 1 | | 29 | 1 | | 30 | 1 | | 31 | 1 | | 32 | 1 | | 35 | 1 | | 37 | 1 | | 38 | 3 | | 40 | 1 | | 42 | 1 | | 43 | 1 | | 44 | 1 | | 45 | 1 | | 46 | 1 | | 47 | 1 | | 48 | 1 | | 49 | 1 | | 50 | 1 | 3. **Step b: Calculate Sample Mean, Median, and Mode** - Mean formula: $$\bar{x} = \frac{\sum f_i x_i}{n}$$ where $f_i$ is frequency, $x_i$ is score, $n=20$. - Calculate sum of all scores: $27+30+45+47+50+38+29+42+31+46+48+43+38+49+38+35+37+32+44+40=800$ - Mean: $$\bar{x} = \frac{800}{20} = 40$$ - Median: Arrange scores in ascending order: 27, 29, 30, 31, 32, 35, 37, 38, 38, 38, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50 - Median position: $$\frac{n+1}{2} = \frac{20+1}{2} = 10.5$$, average of 10th and 11th scores. - 10th score = 38, 11th score = 40 - Median: $$Md = \frac{38 + 40}{2} = 39$$ - Mode: Score with highest frequency is 38 (appears 3 times). - Mode: $$Mo = 38$$ 4. **Step c: Variance and Standard Deviation** - Variance formula for sample: $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$$ - Calculate squared deviations: - $(27-40)^2=169$ - $(29-40)^2=121$ - $(30-40)^2=100$ - $(31-40)^2=81$ - $(32-40)^2=64$ - $(35-40)^2=25$ - $(37-40)^2=9$ - $(38-40)^2=4$ (3 times, total $12$) - $(40-40)^2=0$ - $(42-40)^2=4$ - $(43-40)^2=9$ - $(44-40)^2=16$ - $(45-40)^2=25$ - $(46-40)^2=36$ - $(47-40)^2=49$ - $(48-40)^2=64$ - $(49-40)^2=81$ - $(50-40)^2=100$ - Sum of squared deviations: $169+121+100+81+64+25+9+12+0+4+9+16+25+36+49+64+81+100=965$ - Variance: $$s^2 = \frac{965}{20-1} = \frac{965}{19} \approx 50.79$$ - Standard deviation: $$s = \sqrt{50.79} \approx 7.13$$ **Final answers:** - Mean = 40 - Median = 39 - Mode = 38 - Variance $\approx 50.79$ - Standard deviation $\approx 7.13$