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 \( \text{Range}=\text{Max}-\text{Min} \). Since it directly depends on the maximum and minimum values (extreme values), a single outlier can greatly affect it. For example, if a data - set has values \( \{1,2,3,4,100\} \), the range is \( 100 - 1=99 \), and the outlier \( 100 \) has a huge impact.
• 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 make \( (x_{i}-\bar{x})^{2} \) large. So, extreme values have a significant impact on 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). Quartiles divide the data into four equal parts. Extreme values (either very large or very small) do not affect the calculation of quartiles as long as they are not in the middle 50% of the data. For example, if we have a data - set \( \{1,2,3,4,100\} \), \( Q_{1}=2 \), \( Q_{3}=4 \), and \( \text{IQR}=4 - 2 = 2 \), and the outlier \( 100 \) does not affect the IQR.

Answer:

The IQR is the most resistant to extreme values.