To solve for the quartiles, interquartile range, and fences, we need to follow a series of steps. Let's break it down:

1. Organize the Data in Ascending Order

First, the data must be sorted in ascending order:

225.5, 260.1, 273.7, 288.1, 288.5, 288.7, 308.1, 332.7, 369.2, 378.6, 383.1, 386.3, 405.3, 405.3, 405.3, 407.9, 408.4, 415.5, 425.7, 427, 434.8, 450.2, 459.5, 459.7, 460.5, 475.3, 475.8, 475.8, 476.5, 487.4, 489.1, 490.7, 515.2, 526.7, 528.2, 529.7, 532.1, 538.7, 539.1, 543.6, 543.6, 543.6, 550.2, 550.5, 568, 575.5, 582.8, 587.6, 588.4, 591.6, 626.2, 641.8, 651, 654.2.

2. Find the First Quartile (Q1)

The first quartile (Q1) is the median of the lower half of the data set (excluding the median if the number of data points is odd). The data set has 56 data points, so we split the data into two halves.

• Lower half (first 28 values):
  225.5, 260.1, 273.7, 288.1, 288.5, 288.7, 308.1, 332.7, 369.2, 378.6, 383.1, 386.3, 405.3, 405.3, 405.3, 407.9, 408.4, 415.5, 425.7, 427, 434.8, 450.2, 459.5, 459.7, 460.5, 475.3, 475.8, 475.8, 476.5.

• Median of the lower half (Q1) is the average of the 14th and 15th values, which are 405.3 and 405.3.
  Thus, Q1 = 405.3.

3. Find the Third Quartile (Q3)

The third quartile (Q3) is the median of the upper half of the data set. The upper half (last 28 values) is:

• Upper half (last 28 values):
  475.3, 475.8, 475.8, 476.5, 487.4, 489.1, 490.7, 515.2, 526.7, 528.2, 529.7, 532.1, 538.7, 539.1, 543.6, 543.6, 543.6, 550.2, 550.5, 568, 575.5, 582.8, 587.6, 588.4, 591.6, 626.2, 641.8, 651, 654.2.

• Median of the upper half (Q3) is the average of the 14th and 15th values, which are 543.6 and 543.6.
  Thus, Q3 = 543.6.

4. Find the Interquartile Range (IQR)

The interquartile range (IQR) is the difference between the third quartile and the first quartile: $IQR = Q3 - Q1 = 543.6 - 405.3 = 138.3$

5. Find the Lower Fence (Fencelow)

The lower fence is given by: $Fencelow = Q1 - 1.5 \times IQR = 405.3 - 1.5 \times 138.3 = 405.3 - 207.45 = 197.85$

6. Find the Upper Fence (Fencehigh)

The upper fence is given by: $Fencehigh = Q3 + 1.5 \times IQR = 543.6 + 1.5 \times 138.3 = 543.6 + 207.45 = 751.05$

Summary:

• First Quartile (Q1) = 405.3
• Third Quartile (Q3) = 543.6
• Interquartile Range (IQR) = 138.3
• Lower Fence (Fencelow) = 197.85
• Upper Fence (Fencehigh) = 751.05

Let me know if you'd like further clarification or help with another question!

Follow-up Questions:

1. How do we interpret the lower and upper fences in terms of outliers?
2. What does the interquartile range tell us about the data's spread?
3. How would you calculate quartiles if the data had an odd number of values?
4. How does the IQR method help identify outliers in a data set?
5. What are some other ways to measure the spread of data besides the IQR?

Tip: Always ensure the data is sorted before calculating quartiles to avoid errors.