Question: Here is a sample data set.
370.2 453.9 459.8 513.2 443.9 463.7
496.6 415.4 417.2 508.2 504.8 460.3
393.9 398.3 486.2 350.8 470 415.4
452.5 446 516.1 405.6 355.5 487.5
430.5 517.2 478.7 424.6 430 441.7
456.2 346.3 457 430.4 478.7 478.7
444.5 415.4 435.2 439.2 424 523
320.5 488.7 370 340.5 486.5 409.1
432.2 393 468 495.2 444.5 446
Use a calculator/technology to answer the following.
Find the first quartile for this data set.
Q₁ = 398.3
Find the third quartile for this data set.
Q₃ =
Find the interquartile range for this data set.
IQR = Q₃ − Q₁ =
1. **State the problem:** We are given a data set and asked to find the third quartile $Q_3$ and the interquartile range $IQR$ given that the first quartile $Q_1 = 398.3$.
2. **Recall definitions:**
- The first quartile $Q_1$ is the median of the lower half of the data.
- The third quartile $Q_3$ is the median of the upper half of the data.
- The interquartile range $IQR$ is defined as $$IQR = Q_3 - Q_1$$
3. **Organize the data:**
Sort the data in ascending order (already done or use technology). The data has 54 values.
4. **Find $Q_3$:**
Since $n=54$, the position of $Q_3$ is at the 75th percentile.
The formula for the position of $Q_3$ is $$Q_3 = \text{value at } \left(\frac{3(n+1)}{4}\right)^{th} \text{ position} = \frac{3(54+1)}{4} = \frac{3 \times 55}{4} = 41.25$$
5. **Calculate $Q_3$ value:**
The 41st and 42nd sorted values are needed to interpolate.
From the sorted data, the 41st value is 478.7 and the 42nd value is 478.7.
So,
$$Q_3 = 478.7 + 0.25 \times (478.7 - 478.7) = 478.7$$
6. **Calculate $IQR$:**
$$IQR = Q_3 - Q_1 = 478.7 - 398.3 = 80.4$$
7. **Final answers:**
- Third quartile $Q_3 = 478.7$
- Interquartile range $IQR = 80.4$