1. **State the problem:** We are given a grouped frequency distribution of interest paid to 460 investors and need to find the standard deviation. 2. **Data given:** Interest intervals (ksh'000): 25-29, 30-40, 41-61, 62-82, 83-111 Frequencies: 17, 55, 142, 153, 93 3. **Step 1: Find the midpoints ($x_i$) of each class interval:** $$\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$$ - 25-29: $\frac{25+29}{2} = 27$ - 30-40: $\frac{30+40}{2} = 35$ - 41-61: $\frac{41+61}{2} = 51$ - 62-82: $\frac{62+82}{2} = 72$ - 83-111: $\frac{83+111}{2} = 97$ 4. **Step 2: Calculate the mean ($\bar{x}$):** $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Calculate $\sum f_i x_i$: $$17 \times 27 = 459$$ $$55 \times 35 = 1925$$ $$142 \times 51 = 7242$$ $$153 \times 72 = 11016$$ $$93 \times 97 = 9021$$ Sum: $$459 + 1925 + 7242 + 11016 + 9021 = 29663$$ Total frequency $\sum f_i = 460$ Mean: $$\bar{x} = \frac{29663}{460} \approx 64.48$$ 5. **Step 3: Calculate variance ($\sigma^2$):** Formula: $$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$$ Calculate each squared deviation and multiply by frequency: - For 27: $$ (27 - 64.48)^2 = ( -37.48 )^2 = 1404.35$$ $$17 \times 1404.35 = 23873.95$$ - For 35: $$ (35 - 64.48)^2 = ( -29.48 )^2 = 869.04$$ $$55 \times 869.04 = 47797.2$$ - For 51: $$ (51 - 64.48)^2 = ( -13.48 )^2 = 181.68$$ $$142 \times 181.68 = 25783.56$$ - For 72: $$ (72 - 64.48)^2 = 7.52^2 = 56.54$$ $$153 \times 56.54 = 8647.62$$ - For 97: $$ (97 - 64.48)^2 = 32.52^2 = 1057.63$$ $$93 \times 1057.63 = 98359.59$$ Sum of $f_i (x_i - \bar{x})^2$: $$23873.95 + 47797.2 + 25783.56 + 8647.62 + 98359.59 = 204461.92$$ Variance: $$\sigma^2 = \frac{204461.92}{460} \approx 444.48$$ 6. **Step 4: Calculate standard deviation ($\sigma$):** $$\sigma = \sqrt{444.48} \approx 21.08$$ **Final answer:** The standard deviation of the distribution is approximately **21.08** ksh'000.