1. **Stating the problem:** We have the puzzle solving times in seconds: 548, 490, 856, 420, 534, 470, 632, 597, 473, 450. We need to find the lower quartile (Q1), upper quartile (Q3), interquartile range (IQR), and identify outliers using the formulas: $$Q1 - 1.5 \times IQR$$ and $$Q3 + 1.5 \times IQR$$ 2. **Sort the data:** $$420, 450, 470, 473, 490, 534, 548, 597, 632, 856$$ 3. **Find the lower quartile (Q1):** Q1 is the median of the lower half (first 5 numbers): 420, 450, 470, 473, 490 Median of these is the 3rd value: $$Q1 = 470$$ 4. **Find the upper quartile (Q3):** Q3 is the median of the upper half (last 5 numbers): 534, 548, 597, 632, 856 Median of these is the 3rd value: $$Q3 = 597$$ 5. **Calculate the interquartile range (IQR):** $$IQR = Q3 - Q1 = 597 - 470 = 127$$ 6. **Calculate the outlier boundaries:** Lower boundary: $$Q1 - 1.5 \times IQR = 470 - 1.5 \times 127 = 470 - 190.5 = 279.5$$ Upper boundary: $$Q3 + 1.5 \times IQR = 597 + 1.5 \times 127 = 597 + 190.5 = 787.5$$ 7. **Identify outliers:** Any data point less than 279.5 or greater than 787.5 is an outlier. From the data: - Below 279.5: None - Above 787.5: 856 So, the outlier is 856 seconds. **Final answers:** - Lower quartile (Q1): 470 seconds - Upper quartile (Q3): 597 seconds - Interquartile range (IQR): 127 seconds - Outliers: 856 seconds