Which one of the following statements is correct?

A standard deviation is a measure of the amount of variation or dispersion of a set of values. By definition, the standard deviation is calculated as the square root of the variance, which is the average of the squared differences from the mean. Since squaring any number (whether positive or negative) always results in a non-negative value, taking the average of these squares will also yield a non-negative result. Consequently, when the square root is taken, the standard deviation can only be zero or a positive value. As such, it is indeed impossible to have a sample of measurements with a negative standard deviation.

The other statements touch on different aspects of standard deviation and mean. For instance, while it's theoretically possible to have a standard deviation equal to zero if all measurements are identical, stating that it is impossible suggests a misunderstanding of data sets. Similarly, the idea that the mean cannot be less than the standard deviation is not a universal truth—while it can happen in some distributions, there is no rule prohibiting it in others. τέλος, a positive standard deviation is possible and common in many datasets. Focusing on the properties of standard deviation clarifies why the first statement is correct.