Why is the $standard deviation$ generally considered a better measure of variation than the $range$?

• A. Because the $standard deviation$ is always smaller than the $range$ for any data set.
• B. Because the $standard deviation$ uses all data values and reflects how each value deviates from the $mean$, while the $range$ only considers the smallest and largest values.
• C. Because the $standard deviation$ ignores outliers, while the $range$ is affected by them.
• D. Because the $standard deviation$ is easier to calculate than the $range$.

Answer: (B)

Verified step by step guidance

1. Understand the definition of the range: it is the difference between the maximum and minimum values in a data set, calculated as \(\text{Range} = \text{Max} - \text{Min}\).
2. Recognize that the range only considers two data points (the smallest and largest), which means it does not take into account how the other data values are distributed.
3. Understand the definition of standard deviation: it measures the average amount by which each data value deviates from the mean, using all data points in the calculation.
4. Recall the formula for the standard deviation of a sample: \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\), where \(x_i\) are the data values, \(\bar{x}\) is the sample mean, and \(n\) is the number of data points.
5. Conclude that because the standard deviation incorporates every data value and their deviations from the mean, it provides a more comprehensive and reliable measure of variation than the range, which only reflects the spread between two extreme values.