1. **State the problem:** We have a frequency distribution of low-power FM radio stations in 30 states grouped into class intervals. We need to find the variance and standard deviation of the data. 2. **Identify the class midpoints:** For each class interval, calculate the midpoint $x_i$ as the average of the lower and upper limits. - 1–9: $\frac{1+9}{2} = 5$ - 10–18: $\frac{10+18}{2} = 14$ - 19–27: $\frac{19+27}{2} = 23$ - 28–36: $\frac{28+36}{2} = 32$ - 37–45: $\frac{37+45}{2} = 41$ - 46–54: $\frac{46+54}{2} = 50$ 3. **List frequencies $f_i$:** - 1–9: 5 - 10–18: 7 - 19–27: 10 - 28–36: 3 - 37–45: 3 - 46–54: 2 4. **Calculate the mean $\bar{x}$:** $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{5\times5 + 7\times14 + 10\times23 + 3\times32 + 3\times41 + 2\times50}{30}$$ Calculate numerator: $$5\times5=25,\quad 7\times14=98,\quad 10\times23=230,\quad 3\times32=96,\quad 3\times41=123,\quad 2\times50=100$$ Sum numerator: $$25 + 98 + 230 + 96 + 123 + 100 = 672$$ Mean: $$\bar{x} = \frac{672}{30} = 22.4$$ 5. **Calculate variance $\sigma^2$:** Use formula: $$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$$ Calculate each squared deviation and multiply by frequency: - $(5 - 22.4)^2 = 302.76$, $5 \times 302.76 = 1513.8$ - $(14 - 22.4)^2 = 70.56$, $7 \times 70.56 = 493.92$ - $(23 - 22.4)^2 = 0.36$, $10 \times 0.36 = 3.6$ - $(32 - 22.4)^2 = 92.16$, $3 \times 92.16 = 276.48$ - $(41 - 22.4)^2 = 345.96$, $3 \times 345.96 = 1037.88$ - $(50 - 22.4)^2 = 756.96$, $2 \times 756.96 = 1513.92$ Sum these: $$1513.8 + 493.92 + 3.6 + 276.48 + 1037.88 + 1513.92 = 4839.6$$ Variance: $$\sigma^2 = \frac{4839.6}{30} = 161.32$$ 6. **Calculate standard deviation $\sigma$:** $$\sigma = \sqrt{161.32} \approx 12.7$$ **Final answers:** - Variance = $161.32$ - Standard deviation = $12.7$