Let's calculate the measures of variability for the given data set step by step.
1. Order the Values:
The ordered data set is:
[tex]\[
5, 17, 18, 20, 20, 21, 23, 26, 28, 29
\][/tex]
2. Median:
The middle value of the ordered data set (for 10 values) is the average of the 5th and 6th values.
[tex]\[
\frac{20 + 21}{2} = \frac{41}{2} = 20.5
\][/tex]
3. Range:
The range is the difference between the maximum and minimum values in the ordered data set.
[tex]\[
\text{Range} = 29 - 5 = 24
\][/tex]
So, the range is [tex]\(24\)[/tex] touchdowns.
4. Interquartile Range (IQR):
The interquartile range is the difference between the first quartile (Q1) and the third quartile (Q3).
- The first quartile ([tex]\(Q1\)[/tex]) is the median of the first half of the ordered data, excluding the overall median. The first half is:
[tex]\[
5, 17, 18, 20, 20
\][/tex]
For these 5 values, the median (Q1) is the 3rd value:
[tex]\[
Q1 = 18.5
\][/tex]
- The third quartile ([tex]\(Q3\)[/tex]) is the median of the second half of the ordered data, excluding the overall median. The second half is:
[tex]\[
21, 23, 26, 28, 29
\][/tex]
For these 5 values, the median (Q3) is the 3rd value:
[tex]\[
Q3 = 25.25
\][/tex]
- The interquartile range (IQR) is calculated as:
[tex]\[
\text{IQR} = Q3 - Q1 = 25.25 - 18.5 = 6.75
\][/tex]
Therefore:
- The range is [tex]\(24\)[/tex] touchdowns.
- The interquartile range is [tex]\(6.75\)[/tex] touchdowns.
