1. **Stating the problem:** We want to find which quartile a given number belongs to in a data set. 2. **Understanding quartiles:** Quartiles divide data into four equal parts. The first quartile ($Q_1$) is the 25th percentile, the second quartile ($Q_2$) is the median or 50th percentile, and the third quartile ($Q_3$) is the 75th percentile. 3. **Formula and rules:** To find quartiles, first sort the data in ascending order. Then use the position formula for quartiles: $$Q_k = \frac{k(n+1)}{4}$$ where $k$ is the quartile number (1, 2, or 3) and $n$ is the number of data points. 4. **Finding the quartile of a number:** - Sort the data. - Calculate $Q_1$, $Q_2$, and $Q_3$ values. - Compare the given number to these quartile values: - If the number is less than or equal to $Q_1$, it is in the first quartile. - If it is greater than $Q_1$ but less than or equal to $Q_2$, it is in the second quartile. - If it is greater than $Q_2$ but less than or equal to $Q_3$, it is in the third quartile. - If it is greater than $Q_3$, it is in the fourth quartile. 5. **Example:** Suppose the sorted data is $[2, 4, 7, 10, 12, 15, 18, 20]$ and we want to find the quartile of the number 10. - $n=8$ - Calculate positions: - $Q_1 = \frac{1(8+1)}{4} = 2.25$ (between 2nd and 3rd data points) - $Q_2 = \frac{2(8+1)}{4} = 4.5$ (between 4th and 5th data points) - $Q_3 = \frac{3(8+1)}{4} = 6.75$ (between 6th and 7th data points) - Interpolating values: - $Q_1 \approx 4 + 0.25(7-4) = 4.75$ - $Q_2 \approx 10 + 0.5(12-10) = 11$ - $Q_3 \approx 15 + 0.75(18-15) = 17.25$ - Since 10 is greater than $Q_1=4.75$ but less than or equal to $Q_2=11$, the number 10 is in the **second quartile**. This method helps you determine the quartile placement of any number in a data set.