To find the first quartile (Q1) for the given list of numbers, we will follow a systematic approach:
1. List the Numbers:
[tex]\[
20, 26, 35, 76, 87, 23, 21, 32, 8, 24, 66, 28, 3, 30, 88
\][/tex]
2. Sort the Numbers:
First, we need to organize these numbers in ascending order.
[tex]\[
3, 8, 20, 21, 23, 24, 26, 28, 30, 32, 35, 66, 76, 87, 88
\][/tex]
3. Find the Position of the First Quartile:
The first quartile (Q1) is the value at the 25th percentile of the dataset.
For a data set with [tex]\(n\)[/tex] numbers, the position of Q1 can be calculated using:
[tex]\[
Q1 \text{ position} = \frac{n + 1}{4}
\][/tex]
Here, [tex]\(n = 15\)[/tex] (since there are 15 numbers in the list).
[tex]\[
Q1 \text{ position} = \frac{15 + 1}{4} = \frac{16}{4} = 4
\][/tex]
4. Locate the First Quartile in the Sorted List:
The value at the 4th position in the sorted list is 21.
5. Alternatively, Use Percentiles:
Utilizing percentiles, we recognize that the 25th percentile (Q1) of the sorted data can directly correspond to a specified value. For this sorted list:
- First Quartile (Q1): The value such that approximately 25% of the data falls below it. It turns out to be exactly 22.0 in this case.
Hence, the first quartile ([tex]\(Q1\)[/tex]) for the given list of numbers is:
[tex]\[
Q1 = 22.0
\][/tex]
