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

• A. The data values are all positive.
• B. All the data values are equal to each other.
• C. There are at least two different values in the data set.
• D. The mean of the data is $0$.

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; all data points are exactly the same.
3. Recognize that if all data values are equal, then the difference between each data value and the mean is zero, which leads to a standard deviation of zero.
4. Note that the data values being all positive is not necessary for the standard deviation to be zero; they could be any constant value, positive, negative, or zero.
5. Conclude that the only statement that must be true when the standard deviation is zero is that all the data values are equal to each other.