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

• A. All the data values are equal to each other.
• B. There are no data values in the set.
• C. The data values are all positive.
• D. The mean of the data is $0$.

Answer: (A)

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; in other words, every data point is exactly the same as the mean.
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 having a standard deviation of zero does not imply anything about the number of data values (the set can have one or more values), nor does it imply that the values must be positive or that the mean must be 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.