1. **State the problem:** We are given a distribution with values of $X$ and their corresponding frequencies (heights of bars in the histogram). We need to find: - The total number of observations $n$ - The sum of all $X$ values, $\sum x$ - The sum of the squares of all $X$ values, $\sum x^2$ 2. **Given data:** - $X$ values: 1, 2, 3, 4, 5, 6 - Frequencies: 1, 3, 3, 4, 2, 1 respectively - Total $n = 14$ (given) 3. **Calculate $\sum x$:** Use the formula $\sum x = \sum (x_i \times f_i)$ where $f_i$ is the frequency of $x_i$. $$\sum x = 1 \times 1 + 2 \times 3 + 3 \times 3 + 4 \times 4 + 5 \times 2 + 6 \times 1$$ Calculate step-by-step: $$\sum x = 1 + 6 + 9 + 16 + 10 + 6 = 48$$ 4. **Calculate $\sum x^2$:** Use the formula $\sum x^2 = \sum (x_i^2 \times f_i)$. $$\sum x^2 = 1^2 \times 1 + 2^2 \times 3 + 3^2 \times 3 + 4^2 \times 4 + 5^2 \times 2 + 6^2 \times 1$$ Calculate step-by-step: $$\sum x^2 = 1 \times 1 + 4 \times 3 + 9 \times 3 + 16 \times 4 + 25 \times 2 + 36 \times 1$$ $$\sum x^2 = 1 + 12 + 27 + 64 + 50 + 36 = 190$$ 5. **Summary:** - Total number of observations $n = 14$ - Sum of $X$ values $\sum x = 48$ - Sum of squares $\sum x^2 = 190$