Question

select the correct answer.

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

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

• the range and the iqr will both stay the same.
• the range will stay the same, but the iqr will change.
• the range and the iqr will both change.
• the iqr will stay the same, but the range will change.

Explanation:

Order and analyze the full data set

The given data set of ages is:

$$ \{29, 34, 31, 39, 43, 28, 37, 35, 33, 60, 26, 33, 38, 36, 41\} $$

Sorting the \(n = 15\) data values in ascending order:

$$ 26, 28, 29, 31, 33, 33, 34, 35, 36, 37, 38, 39, 41, 43, 60 $$

The minimum value is \(26\), the maximum value is \(60\), and the outlier is \(60\).

Calculate range and IQR with the outlier

Using the sorted list of \(15\) values:

• Range:

$$ \text{Range} = \text{Max} - \text{Min} = 60 - 26 = 34 $$

• Quartiles:

The median (\(Q_2\)) is the 8th value: \(35\).
The lower half is the first 7 values: \(26, 28, 29, 31, 33, 33, 34\).
The first quartile (\(Q_1\)) is the median of this lower half: \(31\).
The upper half is the last 7 values: \(36, 37, 38, 39, 41, 43, 60\).
The third quartile (\(Q_3\)) is the median of this upper half: \(39\).

• Interquartile Range (IQR):

$$ \text{IQR} = Q_3 - Q_1 = 39 - 31 = 8 $$

Calculate range and IQR without the outlier

Removing the outlier \(60\) leaves \(n = 14\) values:

$$ 26, 28, 29, 31, 33, 33, 34, 35, 36, 37, 38, 39, 41, 43 $$

• Range:

$$ \text{Range} = \text{Max} - \text{Min} = 43 - 26 = 17 $$

• Quartiles:

The median is the average of the 7th and 8th values: \(\frac{34 + 35}{2} = 34.5\).
The lower half is the first 7 values: \(26, 28, 29, 31, 33, 33, 34\).
The first quartile (\(Q_1\)) is the median of this lower half: \(31\).
The upper half is the last 7 values: \(35, 36, 37, 38, 39, 41, 43\).
The third quartile (\(Q_3\)) is the median of this upper half: \(38\).

• Interquartile Range (IQR):

$$ \text{IQR} = Q_3 - Q_1 = 38 - 31 = 7 $$

Compare the results

• The range changed from \(34\) to \(17\).
• The IQR changed from \(8\) to \(7\).
• Therefore, both the range and the IQR will change.

Answer:

• The range and the IQR will both stay the same.
• The range will stay the same, but the IQR will change.
• The range and the IQR will both change. (Correct answer)
• The IQR will stay the same, but the range will change.