Which of the following statements about the standard deviation of a dataset is correct?

• A. The standard deviation is always negative for datasets with a mean less than $0$.
• B. The standard deviation is negative when all data values are the same.
• C. The standard deviation can never be negative.
• D. The standard deviation can be negative if the data contains negative values.

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 dataset.
2. Understand that 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, and taking the square root of a non-negative number results in a non-negative value.
4. Therefore, the standard deviation itself can never be negative, regardless of the values in the dataset or the mean.
5. Conclude that statements suggesting the standard deviation can be negative are incorrect, and the correct statement is: 'The standard deviation can never be negative.'