Question

the five number summary of a dataset was found to be:

45, 49, 56, 60, 67

an observation is considered an outlier if it is below:

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an observation is considered an outlier if it is above:

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Explanation:

Step1: Identify Q1, Q2, Q3

The five - number summary is \( \text{Minimum}, Q_1, \text{Median}(Q_2), Q_3, \text{Maximum} \). Given the five - number summary \( 45, 49, 56, 60, 67 \), we have \( Q_1 = 49 \), \( Q_3=60 \).

Step2: Calculate the Inter - Quartile Range (IQR)

The formula for the inter - quartile range is \( IQR=Q_3 - Q_1 \).
Substitute \( Q_1 = 49 \) and \( Q_3 = 60 \) into the formula: \( IQR=60 - 49=11 \).

Step3: Calculate the lower bound for outliers

The formula for the lower bound (the value below which a data point is considered an outlier) is \( Q_1-1.5\times IQR \).
Substitute \( Q_1 = 49 \) and \( IQR = 11 \): \( 49-1.5\times11=49 - 16.5 = 32.5 \).

Step4: Calculate the upper bound for outliers

The formula for the upper bound (the value above which a data point is considered an outlier) is \( Q_3 + 1.5\times IQR \).
Substitute \( Q_3=60 \) and \( IQR = 11 \): \( 60+1.5\times11=60 + 16.5=76.5 \).

Answer:

An observation is considered an outlier if it is below: \( 32.5 \)

An observation is considered an outlier if it is above: \( 76.5 \)