1. **Problem Statement:** Calculate the mean number of children per family, the mean absolute deviation, and the standard deviation from the given data. 2. **Given Data:** Number of children ($x$): 0, 1, 2, 3, 4, 5, 6, 7, 8 Number of families ($f$): 1, 9, 26, 59, 72, 52, 29, 7, 1 3. **Step 1: Calculate the mean ($\bar{x}$)** Formula: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Calculate total families: $$N = 1+9+26+59+72+52+29+7+1 = 256$$ Calculate $\sum f_i x_i$: $$0\times1 + 1\times9 + 2\times26 + 3\times59 + 4\times72 + 5\times52 + 6\times29 + 7\times7 + 8\times1 = 0 + 9 + 52 + 177 + 288 + 260 + 174 + 49 + 8 = 1017$$ Mean: $$\bar{x} = \frac{1017}{256} \approx 3.9766$$ 4. **Step 2: Calculate the mean absolute deviation (MAD)** Formula: $$\text{MAD} = \frac{\sum f_i |x_i - \bar{x}|}{N}$$ Calculate $|x_i - \bar{x}|$ for each $x_i$: $|0 - 3.9766|=3.9766$, $|1 - 3.9766|=2.9766$, $|2 - 3.9766|=1.9766$, $|3 - 3.9766|=0.9766$, $|4 - 3.9766|=0.0234$, $|5 - 3.9766|=1.0234$, $|6 - 3.9766|=2.0234$, $|7 - 3.9766|=3.0234$, $|8 - 3.9766|=4.0234$ Calculate $\sum f_i |x_i - \bar{x}|$: $$1\times3.9766 + 9\times2.9766 + 26\times1.9766 + 59\times0.9766 + 72\times0.0234 + 52\times1.0234 + 29\times2.0234 + 7\times3.0234 + 1\times4.0234 = 3.9766 + 26.7894 + 51.3916 + 57.6314 + 1.6848 + 53.2168 + 58.6786 + 21.1638 + 4.0234 = 278.5564$$ MAD: $$\frac{278.5564}{256} \approx 1.0883$$ 5. **Step 3: Calculate the standard deviation ($\sigma$)** Formula: $$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{N}}$$ Calculate $(x_i - \bar{x})^2$ for each $x_i$: $3.9766^2=15.8131$, $2.9766^2=8.8603$, $1.9766^2=3.9060$, $0.9766^2=0.9537$, $0.0234^2=0.0005$, $1.0234^2=1.0473$, $2.0234^2=4.0940$, $3.0234^2=9.1409$, $4.0234^2=16.1875$ Calculate $\sum f_i (x_i - \bar{x})^2$: $$1\times15.8131 + 9\times8.8603 + 26\times3.9060 + 59\times0.9537 + 72\times0.0005 + 52\times1.0473 + 29\times4.0940 + 7\times9.1409 + 1\times16.1875 = 15.8131 + 79.7427 + 101.556 + 56.969 + 0.036 + 54.460 + 118.726 + 63.986 + 16.1875 = 507.4768$$ Standard deviation: $$\sigma = \sqrt{\frac{507.4768}{256}} = \sqrt{1.9824} \approx 1.4073$$ **Final answers:** (i) Mean number of children per family $\approx 3.98$ (ii) Mean absolute deviation $\approx 1.09$ (iii) Standard deviation $\approx 1.41$