1. **Problem Statement:** Find the Variance, Standard Deviation (SD), and Coefficient of Variation (CV) for the given grouped data with class intervals and frequencies. 2. **Given Data:** Class Intervals: 0-4, 4-8, 8-12, 12-16 Frequencies (f): 12, 24, 23, 18 3. **Step 1: Calculate midpoints (x) for each class interval:** Midpoint $x = \frac{\text{lower limit} + \text{upper limit}}{2}$ - For 0-4: $\frac{0+4}{2} = 2$ - For 4-8: $\frac{4+8}{2} = 6$ - For 8-12: $\frac{8+12}{2} = 10$ - For 12-16: $\frac{12+16}{2} = 14$ 4. **Step 2: Calculate $f \times x$ and $f \times x^2$ for each class:** - $x^2$ values: $2^2=4$, $6^2=36$, $10^2=100$, $14^2=196$ - Multiply by frequencies: - $f x$: $12\times2=24$, $24\times6=144$, $23\times10=230$, $18\times14=252$ - $f x^2$: $12\times4=48$, $24\times36=864$, $23\times100=2300$, $18\times196=3528$ 5. **Step 3: Calculate totals:** - Total frequency $N = 12 + 24 + 23 + 18 = 77$ - Sum of $f x = 24 + 144 + 230 + 252 = 650$ - Sum of $f x^2 = 48 + 864 + 2300 + 3528 = 6740$ 6. **Step 4: Calculate mean ($\bar{x}$):** $$\bar{x} = \frac{\sum f x}{N} = \frac{650}{77} \approx 8.4416$$ 7. **Step 5: Calculate variance ($\sigma^2$):** Formula for variance of grouped data: $$\sigma^2 = \frac{\sum f x^2}{N} - \left(\bar{x}\right)^2$$ Calculate: $$\frac{6740}{77} \approx 87.5325$$ $$\sigma^2 = 87.5325 - (8.4416)^2 = 87.5325 - 71.2635 = 16.269$$ 8. **Step 6: Calculate standard deviation (SD):** $$\sigma = \sqrt{\sigma^2} = \sqrt{16.269} \approx 4.033$$ 9. **Step 7: Calculate coefficient of variation (CV):** Formula: $$CV = \frac{\sigma}{\bar{x}} \times 100 = \frac{4.033}{8.4416} \times 100 \approx 47.76\%$$ **Final answers:** - Variance $\approx 16.269$ - Standard Deviation $\approx 4.033$ - Coefficient of Variation $\approx 47.76\%$