Let's find the first quartile (Q1) for the given list of numbers.
The first quartile is the median of the lower half of a data set. To find it, we need to follow these steps:
1. List the numbers in order:
First, organize the given numbers in ascending order.
Given numbers are: 17, 48, 30, 70, 65, 53, 44, 61, 1, 69, 78, 33, 60, 20, 3.
Arranging these numbers in ascending order:
[tex]\[
1, 3, 17, 20, 30, 33, 44, 48, 53, 60, 61, 65, 69, 70, 78
\][/tex]
2. Identify the positions of the quartiles:
The first quartile (Q1) is the median of the first half of the data.
3. Dividing the list into two halves:
- The median (middle value) of the entire dataset at position [tex]\((n+1)/2\)[/tex], where [tex]\(n\)[/tex] is the number of observations.
- Since we have 15 numbers (an odd number), the middle value is the 8th number.
Thus, we divide the list into:
- Lower half: [tex]\( \{1, 3, 17, 20, 30, 33, 44, 48\} \)[/tex]
- Upper half: [tex]\( \{53, 60, 61, 65, 69, 70, 78\} \)[/tex]
4. Finding the median of the lower half:
- The lower half array is: [tex]\( \{1, 3, 17, 20, 30, 33, 44, 48\} \)[/tex]
- To find the median (Q1) of this half, we take the average of the 4th and 5th numbers because there are 8 numbers (even count).
The 4th number is 20, and the 5th number is 30.
5. Calculating Q1:
[tex]\[
Q1 = \frac{20 + 30}{2} = \frac{50}{2} = 25
\][/tex]
Thus, the first quartile [tex]\( Q1 \)[/tex] for the given list of numbers is 25.0.
