In introductory statistics, can the standard deviation of a data set be negative?

• A. Yes; it is negative when the mean of the data set is negative.
• B. Yes; it is negative when the variance is negative.
• C. No; standard deviation is always greater than or equal to 0.
• D. Yes; it is negative when more than half of the observations are below the mean.

Answer: (C)

Verified step by step guidance

1. Recall the definition of standard deviation: it measures the average amount by which data points differ from the mean of the data set.
2. Understand that the standard deviation is calculated as the square root of the variance, where variance is the average of the squared differences from the mean.
3. Since variance involves squaring differences, it is always non-negative (zero or positive), because squares of real numbers cannot be negative.
4. Taking the square root of a non-negative number (the variance) results in a value that is also non-negative.
5. Therefore, the standard deviation cannot be negative; it is always greater than or equal to zero regardless of the data values or their mean.