Let's walk through the steps to solve this problem correctly.

Step 1: Organize the Data

We start by sorting the data set in ascending order. The data set is:

410.7, 415.6, 397.6, 373.8, 267.7, 408.9, 
427.6, 538, 380.2, 471.3, 532, 472.6, 
414, 401.8, 337, 361.6, 412.6, 522.5, 
472.9, 358.6, 500.9, 498, 274.5, 412.6, 
388.1, 378.9, 512.9, 463.3, 429.4, 575.7, 
391.2, 406.8, 472.6, 315, 501.8, 459.4, 
406.5, 461.5, 257.1, 472.6, 483.8, 371.4, 
572.3, 288.8, 361.5, 419.9, 378.9, 434.9, 
296.6, 378.9, 379, 396.2, 478.3, 446.8

Step 2: Sort the Data

When we arrange the data in ascending order:

257.1, 267.7, 274.5, 288.8, 296.6, 315, 337, 358.6, 361.5, 361.6, 373.8, 378.9, 378.9, 378.9, 379, 380.2, 
388.1, 391.2, 396.2, 397.6, 401.8, 406.5, 406.8, 408.9, 410.7, 412.6, 412.6, 414, 415.6, 419.9, 427.6, 
429.4, 434.9, 446.8, 459.4, 461.5, 463.3, 472.6, 472.6, 472.6, 472.6, 472.9, 478.3, 483.8, 488, 498, 500.9, 
501.8, 512.9, 522.5, 532, 538, 572.3, 575.7

Step 3: Determine Quartiles

To calculate quartiles:

• Q1 (first quartile) is the median of the lower half of the data.
• Q3 (third quartile) is the median of the upper half of the data.

The dataset has 50 numbers, so the median position is the 25th value, which splits the data into two halves.

Q1 Calculation:

The lower half is:

257.1, 267.7, 274.5, 288.8, 296.6, 315, 337, 358.6, 361.5, 361.6, 373.8, 378.9, 378.9, 378.9, 379, 380.2, 
388.1, 391.2, 396.2, 397.6, 401.8, 406.5, 406.8, 408.9, 410.7

There are 25 values, so the median of this lower half is the 13th number, which is 378.9.

Q3 Calculation:

The upper half is:

412.6, 412.6, 414, 415.6, 419.9, 427.6, 429.4, 434.9, 446.8, 459.4, 461.5, 463.3, 472.6, 472.6, 472.6, 472.6, 
472.9, 478.3, 483.8, 488, 498, 500.9, 501.8, 512.9, 522.5, 532, 538, 572.3, 575.7

There are also 25 values, so the median of this upper half is the 13th number, which is 472.6.

Step 4: Calculate the Interquartile Range (IQR)

The Interquartile Range (IQR) is the difference between Q3 and Q1: $IQR = Q3 - Q1 = 472.6 - 378.9 = 93.7$

Final Answers:

• Q1 = 378.9
• Q3 = 472.6
• IQR = 93.7

---

Would you like more details on any of the steps? Or would you like to explore more questions related to quartiles and IQR?

Here are some related questions:

1. How do you calculate percentiles from a data set?
2. What is the significance of the interquartile range in statistics?
3. How does outlier detection work with quartiles and IQR?
4. How do you use quartiles in box plots?
5. What is the relationship between quartiles and standard deviation?

Tip: To avoid mistakes in quartile calculations, always ensure the data is sorted in ascending order first!