If a set of data has a standard deviation of $0$, which of the following must be true about the data values?

• A. The mean of the data is $0$.
• B. All the data values are equal to each other.
• C. The data values form a perfect normal distribution.
• D. The data values are all positive.

Answer: (B)

Verified step by step guidance

1. Recall the definition of standard deviation: it measures the amount of variation or dispersion of a set of data values from their mean.
2. Understand that a standard deviation of 0 means there is no variation in the data values; they do not deviate at all from the mean.
3. Since there is no variation, all data values must be exactly the same, because any difference would create some positive standard deviation.
4. Note that the mean does not have to be 0; it can be any number, but all data points must equal that mean.
5. Recognize that having a standard deviation of 0 does not imply the data forms a normal distribution or that all values are positive; it only means all values are identical.