(i) To find the standard deviation, use the formula:

\(\text{sd}^2 = \frac{\Sigma(x-c)^2}{n} - \left(\frac{\Sigma(x-c)}{n}\right)^2\)

Substitute the given values:

\(\text{sd}^2 = \frac{1957.5}{30} - \left(\frac{234}{30}\right)^2\)

\(\text{sd}^2 = 65.25 - 6.084\)

\(\text{sd}^2 = 59.166\)

\(\text{sd} = \sqrt{59.166} \approx 2.1\)

(ii) Given the mean is 86, use the formula for the mean:

\(\bar{x} = \frac{\Sigma x}{n} = 86\)

\(86 = \frac{234}{30} + c\)

\(86 = 7.8 + c\)

\(c = 86 - 7.8\)

\(c = 78.2\)

We start with the equation \(\Sigma(x - 36) = -60\). Expanding this gives:

\(\Sigma x - 24 \times 36 = -60\)

\(\Sigma x = 24 \times 36 - 60 = 804\)

Next, we use \(\Sigma(x - 36)^2 = 227.76\). Expanding gives:

\(\Sigma x^2 - 2 \times 36 \times \Sigma x + 24 \times 36^2 = 227.76\)

Substitute \(\Sigma x = 804\):

\(\Sigma x^2 - 2 \times 36 \times 804 + 24 \times 1296 = 227.76\)

\(\Sigma x^2 - 57888 + 31104 = 227.76\)

\(\Sigma x^2 = 227.76 + 57888 - 31104 = 27011.76\)

(i) Let \(n\) be the number of children. The mean height is given by:

\(\frac{\Sigma x}{n} = 104.8\)

We know \(\Sigma(x - 100) = 72\), so:

\(\Sigma x - 100n = 72\)

Substitute \(\Sigma x = 104.8n\) into the equation:

\(104.8n - 100n = 72\)

\(4.8n = 72\)

\(n = \frac{72}{4.8} = 15\)

(ii) We need to find \(\Sigma(x - 104.8)^2\). We have:

\(\Sigma(x - 100)^2 = 499.2\)

\(\Sigma(x - 104.8)^2 = \Sigma(x - 100 + 100 - 104.8)^2\)

\(= \Sigma((x - 100) - 4.8)^2\)

Using the expansion \((a - b)^2 = a^2 - 2ab + b^2\), we have:

\(\Sigma(x - 104.8)^2 = \Sigma(x - 100)^2 - 2 \times 4.8 \times \Sigma(x - 100) + 15 \times 4.8^2\)

\(= 499.2 - 2 \times 4.8 \times 72 + 15 \times 4.8^2\)

\(= 499.2 - 691.2 + 345.6\)

\(= 153.6\)

First, calculate the mean \(\bar{x}\) using the formula:

\(\bar{x} = \frac{\Sigma x}{n} = \frac{645}{150} = 4.3\)

Next, use the formula for variance:

\(\text{Variance} = \frac{\Sigma x^2}{n} - \bar{x}^2 = \frac{8287.5}{150} - 4.3^2\)

Calculate \(\frac{8287.5}{150} = 55.25\) and \(4.3^2 = 18.49\).

Thus, the variance is:

\(55.25 - 18.49 = 36.76\)

The standard deviation is:

\(\sqrt{36.76} = 6.063\)

Finally, calculate \(\Sigma (x - \bar{x})^2\) using:

\(\Sigma (x - \bar{x})^2 = n \times \text{Variance} = 150 \times 36.76 = 5514\)

Rounding to the nearest whole number gives 5510.

(i) To find the standard deviation, first determine the number of data points \(n\) using the equation for the mean:

\(\frac{133}{n} + 25 = 28.325\)

Solving for \(n\):

\(\frac{133}{n} = 3.325\)

\(n = 40\)

Next, use the formula for variance with coded data:

\(\frac{3762}{40} - 3.325^2 = 82.99\)

The standard deviation is:

\(\sqrt{82.99} = 9.11\)

(ii) To find \(\Sigma x^2\), use the uncoded variance formula:

\(82.99 = \frac{\Sigma x^2}{40} - 28.325^2\)

\(\Sigma x^2 = (82.99 + 28.325^2) \times 40\)

\(\Sigma x^2 = 35412\)

Alternatively, using the expanded form:

\(\Sigma(x - 25)^2 = \Sigma x^2 - 50\Sigma x + 40 \times 25^2\)

\(\Sigma x^2 = 3762 + 50 \times 1133 + 25000\)

\(\Sigma x^2 = 35412\)