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 ($\bar{x}$)** The mean is given by: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Calculate total families: $$\sum f_i = 1 + 9 + 26 + 59 + 72 + 52 + 29 + 7 + 1 = 256$$ Calculate $\sum f_i x_i$: $$1\times0 + 9\times1 + 26\times2 + 59\times3 + 72\times4 + 52\times5 + 29\times6 + 7\times7 + 1\times8 = 0 + 9 + 52 + 177 + 288 + 260 + 174 + 49 + 8 = 1017$$ Mean: $$\bar{x} = \frac{1017}{256} \approx 3.9766$$ 4. **Step 2: Calculate the mean absolute deviation (MAD)** Formula: $$\text{MAD} = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$$ Calculate $|x_i - \bar{x}|$ and multiply by $f_i$: - $|0 - 3.9766| \times 1 = 3.9766$ - $|1 - 3.9766| \times 9 = 2.9766 \times 9 = 26.7894$ - $|2 - 3.9766| \times 26 = 1.9766 \times 26 = 51.3916$ - $|3 - 3.9766| \times 59 = 0.9766 \times 59 = 57.6154$ - $|4 - 3.9766| \times 72 = 0.0234 \times 72 = 1.6848$ - $|5 - 3.9766| \times 52 = 1.0234 \times 52 = 53.2168$ - $|6 - 3.9766| \times 29 = 2.0234 \times 29 = 58.6786$ - $|7 - 3.9766| \times 7 = 3.0234 \times 7 = 21.1638$ - $|8 - 3.9766| \times 1 = 4.0234$ Sum: $$3.9766 + 26.7894 + 51.3916 + 57.6154 + 1.6848 + 53.2168 + 58.6786 + 21.1638 + 4.0234 = 278.54$$ MAD: $$\frac{278.54}{256} \approx 1.0883$$ 5. **Step 3: Calculate the standard deviation ($\sigma$)** Formula: $$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$$ Calculate $(x_i - \bar{x})^2 \times f_i$: - $(0 - 3.9766)^2 \times 1 = 15.813$ - $(1 - 3.9766)^2 \times 9 = 8.860 \times 9 = 79.74$ - $(2 - 3.9766)^2 \times 26 = 3.907 \times 26 = 101.58$ - $(3 - 3.9766)^2 \times 59 = 0.954 \times 59 = 56.29$ - $(4 - 3.9766)^2 \times 72 = 0.00055 \times 72 = 0.0396$ - $(5 - 3.9766)^2 \times 52 = 1.047 \times 52 = 54.44$ - $(6 - 3.9766)^2 \times 29 = 4.094 \times 29 = 118.73$ - $(7 - 3.9766)^2 \times 7 = 9.141 \times 7 = 63.99$ - $(8 - 3.9766)^2 \times 1 = 16.19$ Sum: $$15.813 + 79.74 + 101.58 + 56.29 + 0.0396 + 54.44 + 118.73 + 63.99 + 16.19 = 506.82$$ Standard deviation: $$\sigma = \sqrt{\frac{506.82}{256}} = \sqrt{1.980} \approx 1.407$$ **Final answers:** - Mean number of children per family: $\approx 3.98$ - Mean absolute deviation: $\approx 1.09$ - Standard deviation: $\approx 1.41$