1. **Stating the problem:** Calculate Karl Pearson's coefficient of correlation ($r$) between age and playing habits using the given data. 2. **Formula:** $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ where $x$ is age, $y$ is playing habits, and $n$ is the number of data points. 3. **Data:** \begin{array}{c|c|c|c|c} \text{Age }(x) & 20 & 21 & 22 & 23 & 24 & 25 \\ \text{Playing }(y) & 400 & 300 & 180 & 196 & 160 & 24 \\ \end{array} 4. **Calculate sums:** $$\sum x = 20 + 21 + 22 + 23 + 24 + 25 = 135$$ $$\sum y = 400 + 300 + 180 + 196 + 160 + 24 = 1260$$ $$\sum x^2 = 20^2 + 21^2 + 22^2 + 23^2 + 24^2 + 25^2 = 400 + 441 + 484 + 529 + 576 + 625 = 3055$$ $$\sum y^2 = 400^2 + 300^2 + 180^2 + 196^2 + 160^2 + 24^2 = 160000 + 90000 + 32400 + 38416 + 25600 + 576 = 306992$$ $$\sum xy = 20\times400 + 21\times300 + 22\times180 + 23\times196 + 24\times160 + 25\times24 = 8000 + 6300 + 3960 + 4508 + 3840 + 600 = 27208$$ 5. **Number of data points:** $n=6$ 6. **Calculate numerator:** $$n\sum xy - \sum x \sum y = 6 \times 27208 - 135 \times 1260 = 163248 - 170100 = -6852$$ 7. **Calculate denominator:** $$\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)} = \sqrt{(6 \times 3055 - 135^2)(6 \times 306992 - 1260^2)}$$ $$= \sqrt{(18330 - 18225)(1841952 - 1587600)} = \sqrt{105 \times 254352} = \sqrt{26606960} \approx 5158.5$$ 8. **Calculate correlation coefficient:** $$r = \frac{-6852}{5158.5} \approx -1.327$$ 9. **Interpretation:** Since $r$ must be between $-1$ and $1$, this value indicates a calculation or data inconsistency. Please verify data or calculations. **Final answer:** Karl Pearson's coefficient of correlation is approximately $-1.327$ (check data for validity).