Question

use the accompanying data set to complete the following actions.
a. find the quartiles.
b. find the interquartile range.
c. identify any outliers.

64 62 54 56 62 57 60 55 62 64 61 61 61 58 76

a. find the quartiles.
the first quartile, \\(q_1\\), is 57.
the second quartile, \\(q_2\\), is 61.
the third quartile, \\(q_3\\), is 62.
(type integers or decimals.)

b. find the interquartile range.
the interquartile range (iqr) is 5. (type an integer or a decimal.)

c. identify any outliers.
choose the correct answer below.
a. there exists at least one outlier in the data set at .
(use a comma to separate answers as needed.)
b. there are no outliers in the data set.

Explanation:

Sort the dataset

Using the Percentiles and Quantiles knowledge point
The given dataset of size \(n = 15\) is:

$$ \{64, 62, 54, 56, 62, 57, 60, 55, 62, 64, 61, 61, 61, 58, 76\} $$

Sorted in ascending order:

$$ 54, 55, 56, 57, 58, 60, 61, 61, 61, 62, 62, 62, 64, 64, 76 $$

Find the quartiles

Using the Percentiles and Quantiles knowledge point

• Median (\(Q_2\)): The 8th value is \(61\).
• First Quartile (\(Q_1\)): The median of the lower half (first 7 values: \(54, 55, 56, 57, 58, 60, 61\)) is the 4th value, which is \(57\).
• Third Quartile (\(Q_3\)): The median of the upper half (last 7 values: \(61, 62, 62, 62, 64, 64, 76\)) is the 12th value overall, which is \(62\).

Calculate the interquartile range

Using the Interquartile Range knowledge point

$$ \text{IQR} = Q_3 - Q_1 = 62 - 57 = 5 $$

Determine the outlier boundaries

Using the Outlier Detection knowledge point

$$
LATEXBLOCK0
$$

Identify any outliers

Using the Outlier Detection knowledge point

• Values below \(49.5\): None.
• Values above \(69.5\): \(76\).
• Therefore, \(76\) is an outlier.

Answer:

• (A) There exists at least one outlier in the data set at 76. (Correct answer)
• (B) There are no outliers in the data set.