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Select the correct answer.

Kara recorded the ages of the people in her book club in this table.

\begin{tabular}{|l|l|l|l|l|}
\hline 29 & 34 & 31 & 39 & 43 \\
\hline 28 & 37 & 35 & 33 & 60 \\
\hline 26 & 33 & 38 & 36 & 41 \\
\hline
\end{tabular}

If the outlier is not included, what will happen to the range and the IQR of this data set?

A. The range and the IQR will both change.
B. The range and the IQR will both stay the same.
C. The range will stay the same, but the IQR will change.
D. The IQR will stay the same, but the range will change.



Answer :

GinnyAnswer
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.

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