If the standard deviation of a set of scores is zero, it means that all the scores in the distribution are the same. This is because the standard deviation measures the amount of variation or dispersion of a set of values. A standard deviation of zero indicates that there is no variation; every score is identical.
For example, consider a set of scores: 85, 85, 85, 85, 85. In this case, every score is 85. When we calculate the standard deviation of these scores, we find that:
- The mean (or average) is 85.
- The differences from the mean are: (85 – 85), (85 – 85), (85 – 85), (85 – 85), (85 – 85), which are all zero.
- Since all differences are zero, the variance (the average of the squared differences) is also zero, and therefore the standard deviation, which is the square root of the variance, is also zero.
This confirms that when the standard deviation is zero, there is no spread in the data; all scores are the same.