First, let's comprehend the concepts of "range" and "interquartile range (IQR)":
1. Range: It is the difference between the highest and lowest values in a data set.
2. Interquartile Range (IQR): It measures the spread of the middle 50% of a data set and is calculated by subtracting the first quartile (Q1) from the third quartile (Q3).
We have Kara’s data set:
[tex]\[
\{29, 34, 31, 39, 43, 28, 37, 35, 33, 60, 26, 33, 38, 36, 41\}
\][/tex]
Let's start by identifying the outlier and removing it for our calculations.
1. Determining and Removing the Outlier:
- The outlier in this data set is 60.
2. Calculate Range and IQR with and without the outlier:
### With the outlier (60 included):
- Range:
- Highest value = 60
- Lowest value = 26
- Range = 60 - 26 = 34
- IQR:
- First Quartile (Q1) ≈ 32.5
- Third Quartile (Q3) ≈ 39
- IQR = 39 - 32.5 = 6.5
### Without the outlier (60 excluded):
- The new data set:
[tex]\[
\{29, 34, 31, 39, 43, 28, 37, 35, 33, 26, 33, 38, 36, 41\}
\][/tex]
- Range:
- Highest value = 43
- Lowest value = 26
- Range = 43 - 26 = 17
- IQR:
- First Quartile (Q1) ≈ 32.25
- Third Quartile (Q3) ≈ 38.5
- IQR = 38.5 - 32.25 = 6.25
### Analyzing the changes:
1. Range:
- Range with outlier: 34
- Range without outlier: 17
- The range changes.
2. IQR:
- IQR with outlier: 6.5
- IQR without outlier: 6.25
- The IQR changes as well, albeit slightly.
Therefore, the correct answer is:
A. The range and the IQR will both change.
