Question

which of the following statements is true about the iqr, range, and standard deviation?
the iqr is the least resistant to extreme values.
the iqr is the most resistant to extreme values.
the range is the most resistant to extreme values.
the standard deviation is the most resistant to extreme values.

Explanation:

• Range: It is calculated as \( \text{Range}=\text{Max}-\text{Min} \). A single extreme value (either very large or very small) can greatly affect the range. For example, if we have a data set \( \{1,2,3,4, 100\} \), the range is \( 100 - 1=99 \), and the value \( 100 \) is an outlier that makes the range large.
• Standard deviation: The formula for the sample standard deviation is \( s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}} \). Extreme values \( x_{i} \) will have large \( (x_{i}-\bar{x})^{2} \) terms. So, extreme values can inflate the standard deviation.
• IQR (Inter - Quartile Range): It is calculated as \( \text{IQR}=Q_{3}-Q_{1} \), where \( Q_{1} \) is the first quartile (25th percentile) and \( Q_{3} \) is the third quartile (75th percentile). The IQR focuses on the middle 50% of the data. Extreme values in the lower 25% or upper 25% of the data do not affect the IQR. For example, if we have a data set \( \{1,2,3,4,100\} \), \( Q_{1} = 2 \), \( Q_{3}=4 \), and \( \text{IQR}=4 - 2=2 \)

Answer:

The IQR is the most resistant to extreme values.