1. **State the problem:** We need to find the mean, variance, and standard deviation for the grouped data given in the table with class limits and frequencies. 2. **Identify class midpoints:** Calculate the midpoint $x_i$ for each class interval by averaging the lower and upper limits. - For 2 - 4: $x_1 = \frac{2 + 4}{2} = 3$ - For 5 - 7: $x_2 = \frac{5 + 7}{2} = 6$ - For 8 - 10: $x_3 = \frac{8 + 10}{2} = 9$ - For 11 - 13: $x_4 = \frac{11 + 13}{2} = 12$ - For 14 - 16: $x_5 = \frac{14 + 16}{2} = 15$ 3. **List frequencies $f_i$:** $f_1=5$, $f_2=9$, $f_3=14$, $f_4=7$, $f_5=5$ 4. **Calculate total frequency $n$:** $$n = 5 + 9 + 14 + 7 + 5 = 40$$ 5. **Calculate the mean $\bar{x}$ using formula:** $$\bar{x} = \frac{\sum f_i x_i}{n}$$ Calculate $f_i x_i$ for each class: - $5 \times 3 = 15$ - $9 \times 6 = 54$ - $14 \times 9 = 126$ - $7 \times 12 = 84$ - $5 \times 15 = 75$ Sum these: $$\sum f_i x_i = 15 + 54 + 126 + 84 + 75 = 354$$ Calculate mean: $$\bar{x} = \frac{354}{40} = 8.85$$ 6. **Calculate variance $\sigma^2$ using formula:** $$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{n}$$ Calculate each $(x_i - \bar{x})^2$: - $(3 - 8.85)^2 = (-5.85)^2 = 34.2225$ - $(6 - 8.85)^2 = (-2.85)^2 = 8.1225$ - $(9 - 8.85)^2 = 0.15^2 = 0.0225$ - $(12 - 8.85)^2 = 3.15^2 = 9.9225$ - $(15 - 8.85)^2 = 6.15^2 = 37.8225$ Multiply by frequencies $f_i$: - $5 \times 34.2225 = 171.1125$ - $9 \times 8.1225 = 73.1025$ - $14 \times 0.0225 = 0.315$ - $7 \times 9.9225 = 69.4575$ - $5 \times 37.8225 = 189.1125$ Sum these: $$\sum f_i (x_i - \bar{x})^2 = 171.1125 + 73.1025 + 0.315 + 69.4575 + 189.1125 = 503.1$$ Calculate variance: $$\sigma^2 = \frac{503.1}{40} = 12.5775$$ 7. **Calculate standard deviation $\sigma$:** $$\sigma = \sqrt{12.5775} \approx 3.547$$ **Final answers:** - Mean $\bar{x} = 8.85$ - Variance $\sigma^2 = 12.5775$ - Standard deviation $\sigma \approx 3.547$