1. **Stating the problem:** You asked where each number in the quartile calculation example comes from. 2. **Data set:** The numbers come from the sorted data set $[2, 4, 7, 10, 12, 15, 18, 20]$. 3. **Number of data points ($n$):** There are 8 numbers in the data set, so $n=8$. 4. **Quartile position formula:** We use the formula $$Q_k = \frac{k(n+1)}{4}$$ where $k$ is the quartile number (1, 2, or 3). 5. **Calculating quartile positions:** - For $Q_1$: $$Q_1 = \frac{1(8+1)}{4} = \frac{9}{4} = 2.25$$ which means the first quartile lies between the 2nd and 3rd data points. - For $Q_2$: $$Q_2 = \frac{2(8+1)}{4} = \frac{18}{4} = 4.5$$ which lies between the 4th and 5th data points. - For $Q_3$: $$Q_3 = \frac{3(8+1)}{4} = \frac{27}{4} = 6.75$$ which lies between the 6th and 7th data points. 6. **Interpolating quartile values:** Since quartile positions are not whole numbers, we interpolate between the data points: - $Q_1$ is between 2nd (4) and 3rd (7) data points: $$4 + 0.25(7-4) = 4.75$$ - $Q_2$ is between 4th (10) and 5th (12) data points: $$10 + 0.5(12-10) = 11$$ - $Q_3$ is between 6th (15) and 7th (18) data points: $$15 + 0.75(18-15) = 17.25$$ 7. **Summary:** Each number comes from the original sorted data or from calculations using the quartile position formula and interpolation between data points.