To find the first quartile $Q_1$, we must first organize the data in ascending order. The quartiles divide the data into four equal parts. The first quartile $Q_1$ is the median of the lower half of the data set, not including the overall median.

Step 1: Arrange the data in ascending order:

1, 6, 13, 21, 31, 50, 66, 181, 414, 635

Step 2: Determine the position of $Q_1$:

The first quartile $Q_1$ is located at $\frac{n+1}{4}$, where $n$ is the number of data points.

• $n = 10$ (number of data points)

So, $\frac{10+1}{4} = \frac{11}{4} = 2.75$. This means $Q_1$ lies between the 2nd and 3rd data points in the ordered set.

Step 3: Calculate $Q_1$:

We can now interpolate between the 2nd and 3rd data points (6 and 13) to find $Q_1$: $Q_1 = 6 + 0.75 \times (13 - 6) = 6 + 0.75 \times 7 = 6 + 5.25 = 11.25$

Thus, the first quartile $Q_1$ is 11.25.

Would you like more details or have any other questions?

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Here are 5 relative questions for further exploration:

1. How do you calculate the third quartile $Q_3$ from the same data set?
2. What is the interquartile range (IQR) for this data?
3. How does finding the median differ from finding quartiles?
4. How would you handle quartile calculation with an odd number of data points?
5. Can you use quartiles to detect outliers in the data?

Tip: Quartiles are useful in summarizing the spread and distribution of data, especially in detecting variability and potential outliers!