To calculate the first and third quartiles of the given data set, we need to follow these steps:

Step 1: Arrange the data in ascending order

The data you provided is: $10, 15, 25, 19, 37, 62, 29, 8, 6, 30, 15, 20$

After sorting in ascending order: $6, 8, 10, 15, 15, 19, 20, 25, 29, 30, 37, 62$

Step 2: Determine the median (Q2)

The median divides the data set into two halves. Since there are 12 data points, we take the average of the 6th and 7th numbers:

• The 6th number is 19.
• The 7th number is 20.

The median (Q2) is: $Q2 = \frac{19 + 20}{2} = 19.5$

Step 3: Calculate the first quartile (Q1)

The first quartile is the median of the lower half of the data (values before Q2). The lower half of the data is: $6, 8, 10, 15, 15, 19$

The median of this subset is the average of the 3rd and 4th numbers:

• The 3rd number is 10.
• The 4th number is 15.

Thus, the first quartile (Q1) is: $Q1 = \frac{10 + 15}{2} = 12.5$

Step 4: Calculate the third quartile (Q3)

The third quartile is the median of the upper half of the data (values after Q2). The upper half of the data is: $20, 25, 29, 30, 37, 62$

The median of this subset is the average of the 3rd and 4th numbers:

• The 3rd number is 29.
• The 4th number is 30.

Thus, the third quartile (Q3) is: $Q3 = \frac{29 + 30}{2} = 29.5$

Final Answer:

• First Quartile (Q1): 12.5
• Third Quartile (Q3): 29.5

Do you want to dive into any further details or have questions on how these are calculated? Here's a related set of questions you might find interesting:

1. How do you find the interquartile range (IQR)?
2. What is the importance of quartiles in data analysis?
3. Can the median also be considered a form of the mean? How do they differ?
4. How do outliers affect the quartiles?
5. What are other methods to find the quartiles in a data set?

Tip: When working with quartiles, make sure the data is sorted correctly, as the position of numbers is key to accurate calculations.