Certainly! Let's find the first quartile (Q1), the second quartile (Q2), and the third quartile (Q3) for the given data using the step-by-step procedure. We have the following distances traveled to attend college:
[tex]\[ 36, 44, 12, 18, 48, 70, 2, 26, 28, 28, 10, 58, 68, 28, 80, 22, 18, 42 \][/tex]
### Step 1: Order the Data
First, we list the data in ascending order:
[tex]\[ 2, 10, 12, 18, 18, 22, 26, 28, 28, 28, 36, 42, 44, 48, 58, 68, 70, 80 \][/tex]
### Step 2: Find the Median (Q2)
The median is the middle value of the ordered dataset. Since there are 18 observations (an even number), the median will be the average of the 9th and 10th values.
The 9th value is 28 and the 10th value is also 28. Thus, the median (Q2) is:
[tex]\[ Q2 = \frac{28 + 28}{2} = 28 \][/tex]
### Step 3: Find the First Quartile (Q1)
The first quartile (Q1) is the median of the first half of the data. Since there are 9 values in the first half:
[tex]\[ 2, 10, 12, 18, 18, 22, 26, 28, 28 \][/tex]
The median of this subset (the 25th percentile) is the 5th value, which is:
[tex]\[ Q1 = 18 \][/tex]
### Step 4: Find the Third Quartile (Q3)
The third quartile (Q3) is the median of the second half of the data. The second half is:
[tex]\[ 28, 36, 42, 44, 48, 58, 68, 70, 80 \][/tex]
The median of this subset (the 75th percentile) is the 5th value, which is:
[tex]\[ Q3 = 48 \][/tex]
### Conclusion
Thus, the quartiles of the data are:
[tex]\[ Q_1 = 19.0 \][/tex]
[tex]\[ Q_2 = 28.0 \][/tex]
[tex]\[ Q_3 = 47.0 \][/tex]
So, the quartiles are:
[tex]\[
\begin{array}{l}
Q_1= 19.0 \\
Q_2= 28.0 \\
Q_3= 47.0
\end{array}
\][/tex]
These values represent the first, second (median), and third quartiles of the distance values respectively.
