1. **Problem statement:** Calculate the coefficient of skewness and kurtosis for the exam scores \{60,62,65,68,70,70,72,74,75,75,76,78,80,82,85,88,90,92,95,98\} and interpret the distribution. 2. **Formulas:** - Coefficient of skewness (using Pearson's moment coefficient): $$\text{Skewness} = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^3}{\left(\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\right)^{3/2}}$$ - Coefficient of kurtosis (excess kurtosis): $$\text{Kurtosis} = \frac{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^4}{\left(\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\right)^2} - 3$$ 3. **Step 1: Calculate the mean \(\bar{x}\):** $$\bar{x} = \frac{60 + 62 + 65 + 68 + 70 + 70 + 72 + 74 + 75 + 75 + 76 + 78 + 80 + 82 + 85 + 88 + 90 + 92 + 95 + 98}{20} = \frac{1440}{20} = 72$$ 4. **Step 2: Calculate deviations \(x_i - \bar{x}\):** \(60-72=-12, 62-72=-10, 65-72=-7, 68-72=-4, 70-72=-2, 70-72=-2, 72-72=0, 74-72=2, 75-72=3, 75-72=3, 76-72=4, 78-72=6, 80-72=8, 82-72=10, 85-72=13, 88-72=16, 90-72=18, 92-72=20, 95-72=23, 98-72=26\) 5. **Step 3: Calculate variance \(s^2\):** $$s^2 = \frac{1}{20} \sum (x_i - 72)^2 = \frac{1444}{20} = 72.2$$ 6. **Step 4: Calculate standard deviation \(s\):** $$s = \sqrt{72.2} \approx 8.495$$ 7. **Step 5: Calculate skewness numerator \(\frac{1}{n}\sum (x_i - \bar{x})^3\):** Sum of cubes of deviations: $$(-12)^3 + (-10)^3 + (-7)^3 + (-4)^3 + (-2)^3 + (-2)^3 + 0^3 + 2^3 + 3^3 + 3^3 + 4^3 + 6^3 + 8^3 + 10^3 + 13^3 + 16^3 + 18^3 + 20^3 + 23^3 + 26^3 = -1728 -1000 -343 -64 -8 -8 + 0 + 8 + 27 + 27 + 64 + 216 + 512 + 1000 + 2197 + 4096 + 5832 + 8000 + 12167 + 17576 = 35479$$ Divide by 20: $$\frac{35479}{20} = 1773.95$$ 8. **Step 6: Calculate skewness:** $$\text{Skewness} = \frac{1773.95}{(72.2)^{3/2}} = \frac{1773.95}{(72.2)^{1.5}} = \frac{1773.95}{612.7} \approx 2.89$$ 9. **Step 7: Calculate kurtosis numerator \(\frac{1}{n}\sum (x_i - \bar{x})^4\):** Sum of fourth powers of deviations: $$(-12)^4 + (-10)^4 + (-7)^4 + (-4)^4 + (-2)^4 + (-2)^4 + 0^4 + 2^4 + 3^4 + 3^4 + 4^4 + 6^4 + 8^4 + 10^4 + 13^4 + 16^4 + 18^4 + 20^4 + 23^4 + 26^4 = 20736 + 10000 + 2401 + 256 + 16 + 16 + 0 + 16 + 81 + 81 + 256 + 1296 + 4096 + 10000 + 28561 + 65536 + 104976 + 160000 + 279841 + 456976 = 1,186,040$$ Divide by 20: $$\frac{1,186,040}{20} = 59,302$$ 10. **Step 8: Calculate kurtosis:** $$\text{Kurtosis} = \frac{59,302}{(72.2)^2} - 3 = \frac{59,302}{5,215.84} - 3 \approx 11.37 - 3 = 8.37$$ 11. **Step 9: Interpretation:** - Skewness \(\approx 2.89\) is positive and large, indicating the distribution is right-skewed (asymmetric with a longer tail on the right). - Kurtosis \(\approx 8.37\) is much greater than 0, indicating a leptokurtic distribution, which is more peaked and has heavier tails than a normal distribution. **Final answers:** - Coefficient of skewness: \(\approx 2.89\) - Coefficient of kurtosis: \(\approx 8.37\) - The distribution is asymmetric (right-skewed) and more peaked than a normal distribution.