1. **Problem statement:** Calculate the mean height, variance, and standard deviation for the heights of 8 students: 165, 170, 190, 180, 175, 185, 176, 184 cm. 2. **Mean height calculation:** The mean height $\bar{h}$ is given by: $$\bar{h} = \frac{\sum h}{n}$$ where $\sum h$ is the sum of all heights and $n$ is the number of students. Sum of heights: $$165 + 170 + 190 + 180 + 175 + 185 + 176 + 184 = 1425$$ Number of students $n = 8$ Mean height: $$\bar{h} = \frac{1425}{8} = 178.125$$ 3. **Variance calculation:** Variance $\sigma^2$ is calculated by: $$\sigma^2 = \frac{\sum h^2}{n} - \left(\bar{h}\right)^2$$ Given $\sum h^2 = 254307$ Calculate: $$\frac{254307}{8} = 31788.375$$ Square of mean: $$(178.125)^2 = 31725.39$$ Variance: $$\sigma^2 = 31788.375 - 31725.39 = 62.985$$ 4. **Standard deviation calculation:** Standard deviation $\sigma$ is the square root of variance: $$\sigma = \sqrt{62.985} \approx 7.94$$ **Final answers:** - Mean height = 178.13 cm - Variance = 62.99 cm$^2$ - Standard deviation = 7.94 cm