To find $Q_1$ (the first quartile), we must follow these steps:

Step 1: Arrange the data in ascending order

The given data set is: $35, 29, 44, 72, 34, 64, 41, 50, 54, 104, 39, 58$

Arranging in ascending order: $29, 34, 35, 39, 41, 44, 50, 54, 58, 64, 72, 104$

Step 2: Find the position of $Q_1$

For a sample of size $n = 12$, the first quartile $Q_1$ is the median of the lower half of the data.

The lower half includes the first 6 data points: $29, 34, 35, 39, 41, 44$

The median of this subset is the average of the 3rd and 4th values (since $n = 6$, an even number): $Q_1 = \frac{35 + 39}{2} = \frac{74}{2} = 37$

Answer:

The first quartile $Q_1$ is 37.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How do you find $Q_3$ from the same dataset?
2. How does the interquartile range (IQR) relate to the quartiles?
3. Can $Q_1$ be calculated using an alternative method?
4. What is the importance of quartiles in statistical analysis?
5. How would you interpret $Q_1$ in this context?

Tip: Quartiles are useful for identifying the spread and skewness of the data, with $Q_1$ representing the 25th percentile.