1. **State the problem:** We are given a frequency distribution table with scores $x$ and their frequencies $f$. We need to find the total number of scores $n$, the sum of all scores $\Sigma X$, and the sum of the squares of the scores $\Sigma X^2$. 2. **Given data:** $$\begin{array}{c|c} x & f \\\hline 15 & 1 \\ 14 & 1 \\ 13 & 2 \\ 12 & 3 \\ 11 & 5 \\ 10 & 4 \\ \end{array}$$ 3. **Calculate $n$ (total number of scores):** $$n = \sum f = 1 + 1 + 2 + 3 + 5 + 4 = 16$$ 4. **Calculate $\Sigma X$ (sum of all scores):** $$\Sigma X = \sum (x \times f) = (15 \times 1) + (14 \times 1) + (13 \times 2) + (12 \times 3) + (11 \times 5) + (10 \times 4)$$ $$= 15 + 14 + 26 + 36 + 55 + 40 = 186$$ 5. **Calculate $\Sigma X^2$ (sum of squares of scores):** $$\Sigma X^2 = \sum (x^2 \times f) = (15^2 \times 1) + (14^2 \times 1) + (13^2 \times 2) + (12^2 \times 3) + (11^2 \times 5) + (10^2 \times 4)$$ $$= (225 \times 1) + (196 \times 1) + (169 \times 2) + (144 \times 3) + (121 \times 5) + (100 \times 4)$$ $$= 225 + 196 + 338 + 432 + 605 + 400 = 2196$$ **Final answers:** - $n = 16$ - $\Sigma X = 186$ - $\Sigma X^2 = 2196$