To understand what it means when a set of test scores has a standard deviation of zero, let’s first recall the definition of standard deviation.
The standard deviation of a data set is a measure of how spread out the numbers in the data set are. Specifically, it quantifies the amount of variation or dispersion of a set of values. The key points to note are:
- A higher standard deviation indicates that the data points are spread out over a wider range of values.
- A lower standard deviation indicates that the data points are closer to the mean (average) of the data set.
Now, when the standard deviation is zero, it indicates no variation in the data set. This means that every value in the data set is exactly the same because there is no spread or deviation from the mean.
To address the given options:
- Option A: "It means that the mean of the scores is exactly in the middle of the range."
This option is incorrect. The mean being in the middle of the range does not necessarily imply that the standard deviation is zero. The scores could still vary around the mean.
- Option B: "It means that the scores have the same spread above and below the mean."
This option is incorrect. If the scores have the same spread above and below the mean, there would still be some variation, and thus, the standard deviation would not be zero.
- Option C: "It means that all the scores are zero."
This option is incorrect. While having all scores as zero would give a standard deviation of zero, it is not the only scenario. The scores could be any identical value, not necessarily zero, for the standard deviation to be zero.
- Option D: "It means that all the scores are the same value."
This option is correct. If all the scores are the same, there is no variation among them, hence the standard deviation would indeed be zero.
Therefore, the correct answer is:
D. It means that all the scores are the same value.
