1. **State the problem:** We have a set of shoe sizes: 5.5, 6, 7, 8.5, 6.5, 6.5, 8, 7.5, 8, 5. We want to find how the Interquartile Range (IQR) changes if we add a shoe size of 7 to this data. 2. **Recall the formula and concept:** The IQR is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$): $$\text{IQR} = Q_3 - Q_1$$ The quartiles divide the data into four equal parts when the data is sorted. 3. **Sort the original data:** $$5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5$$ 4. **Find $Q_1$ and $Q_3$ for original data:** - $Q_1$ is the median of the lower half (first 5 values): 5, 5.5, 6, 6.5, 6.5 Median of these is 6 (middle value). - $Q_3$ is the median of the upper half (last 5 values): 7, 7.5, 8, 8, 8.5 Median of these is 8. 5. **Calculate original IQR:** $$\text{IQR} = 8 - 6 = 2$$ 6. **Add the new data point 7 and sort:** $$5, 5.5, 6, 6.5, 6.5, 7, 7, 7.5, 8, 8, 8.5$$ 7. **Find $Q_1$ and $Q_3$ for new data:** - Number of data points is now 11. - Median is the 6th value: 7. - Lower half (first 5 values): 5, 5.5, 6, 6.5, 6.5 Median is 6 ($Q_1$). - Upper half (last 5 values): 7, 7.5, 8, 8, 8.5 Median is 8 ($Q_3$). 8. **Calculate new IQR:** $$\text{IQR} = 8 - 6 = 2$$ 9. **Conclusion:** The IQR does not change when adding a shoe size of 7. **Final answer:** The IQR remains the same at 2.