To find the first quartile (Q1) for the given list of numbers, we will follow these steps:
1. List out all the numbers:
[tex]\[
71, 91, 8, 25, 18, 81, 36, 85, 57, 31, 1, 47, 51, 9, 20
\][/tex]
2. Arrange the numbers in ascending order:
[tex]\[
1, 8, 9, 18, 20, 25, 31, 36, 47, 51, 57, 71, 81, 85, 91
\][/tex]
3. Identify the position of the first quartile (Q1):
The first quartile is the value that separates the lowest 25% of the data from the rest. To find Q1, we use the formula for the position of a percentile:
[tex]\[
Q1 = \left(\frac{25}{100} \times (N + 1)\right)^{th} \text{ value}
\][/tex]
where [tex]\( N \)[/tex] is the number of data points.
4. Calculate the position:
Given [tex]\( N = 15 \)[/tex] (since there are 15 numbers in the list),
[tex]\[
Q1 = \left(\frac{25}{100} \times (15 + 1)\right)^{th} \text{ value} = \left(\frac{25}{100} \times 16\right)^{th} \text{ value} = 4^{th} \text{ value}
\][/tex]
5. Find the 4th value in the ordered list:
The 4th value in the ordered list is:
[tex]\[
18
\][/tex]
So, the first quartile (Q1) for the given list of numbers is [tex]\( 19.0 \)[/tex].
