Sure, let's find the lower quartile (Q1), median (Q2), and upper quartile (Q3) of the given data set step-by-step.
Given data:
[tex]\[ 4, 6, 8, 12, 14, 16, 20 \][/tex]
### Step 1: Order the data
Ensure that the data is in ascending order. The given data is already sorted:
[tex]\[ 4, 6, 8, 12, 14, 16, 20 \][/tex]
### Step 2: Find the Median (Q2)
The median (Q2) is the middle value of the data. Since we have 7 data points, the median is the 4th value in the ordered list.
[tex]\[ \text{Median (Q2)} = 12 \][/tex]
### Step 3: Find the Lower Quartile (Q1)
The lower quartile (Q1) is the median of the first half of the data. Exclude the median when the total number of data points is odd. So, we take the first half:
[tex]\[ 4, 6, 8 \][/tex]
Since there are 3 data points in the first half, the median (and thus Q1) is the second value:
[tex]\[ \text{Q1} = 6 \][/tex]
### Step 4: Find the Upper Quartile (Q3)
The upper quartile (Q3) is the median of the second half of the data. Again, excluding the overall median, we take the second half:
[tex]\[ 14, 16, 20 \][/tex]
For this subset of 3 data points, the median (and thus Q3) is the second value:
[tex]\[ \text{Q3} = 16 \][/tex]
### Final Results
- Lower Quartile (Q1): 6
- Median (Q2): 12
- Upper Quartile (Q3): 16
So, the lower quartile, median and upper quartile of the given data are [tex]\( 6, 12, \)[/tex] and [tex]\( 16 \)[/tex] respectively.
