Question
What is the third quartile of this data set?
14, 18, 20, 21, 25, 32, 38, 42, 48
38
36
42
40
14, 18, 20, 21, 25, 32, 38, 42, 48
38
36
42
40
Answers
Answered by
GPT-4o mini
To find the third quartile (Q3) of the given data set, we need to first arrange the data in ascending order, which it already is:
14, 18, 20, 21, 25, 32, 38, 42, 48
Next, we'll determine the position of the third quartile. The third quartile is the median of the upper half of the data.
1. **Finding the median position**:
- The median (Q2) divides the data into two halves. Since there are 9 data points (an odd number), the median position is at \( (N + 1) / 2 \) where \( N \) is the total number of data points.
- \( (9 + 1) / 2 = 5 \)
- The median (Q2) is the 5th value in the ordered list, which is 25.
2. **Splitting the data**:
- The lower half of the data (below the median) is: 14, 18, 20, 21 (4 values).
- The upper half of the data (above the median) is: 32, 38, 42, 48 (4 values).
3. **Finding the third quartile (Q3)**:
- To find Q3, we need the median of the upper half: 32, 38, 42, 48.
- Since there are 4 values (an even number), the median is the average of the two middle values.
- The two middle values in the upper half are 38 and 42.
- \( Q3 = (38 + 42) / 2 = 80 / 2 = 40 \).
Thus, the third quartile (Q3) of the data set is \( \boxed{40} \).
14, 18, 20, 21, 25, 32, 38, 42, 48
Next, we'll determine the position of the third quartile. The third quartile is the median of the upper half of the data.
1. **Finding the median position**:
- The median (Q2) divides the data into two halves. Since there are 9 data points (an odd number), the median position is at \( (N + 1) / 2 \) where \( N \) is the total number of data points.
- \( (9 + 1) / 2 = 5 \)
- The median (Q2) is the 5th value in the ordered list, which is 25.
2. **Splitting the data**:
- The lower half of the data (below the median) is: 14, 18, 20, 21 (4 values).
- The upper half of the data (above the median) is: 32, 38, 42, 48 (4 values).
3. **Finding the third quartile (Q3)**:
- To find Q3, we need the median of the upper half: 32, 38, 42, 48.
- Since there are 4 values (an even number), the median is the average of the two middle values.
- The two middle values in the upper half are 38 and 42.
- \( Q3 = (38 + 42) / 2 = 80 / 2 = 40 \).
Thus, the third quartile (Q3) of the data set is \( \boxed{40} \).