If all scores are equal, what is the standard deviation?

If all scores are equal, what is the standard deviation?

• A. Zero.
• B. Equal to the mean.
• C. Undefined.
• D. Equal to the range.

Answer: (A)

The standard deviation measures how far scores tend to be from the average. When every score is the same, there is no variability at all, so all deviations from the mean are zero. Squaring those zeros and averaging them still gives zero, which means the standard deviation is zero. This holds for both population and sample calculations. It’s not equal to the mean, since it reflects spread, not central value. It isn’t undefined, because the usual formulas produce a well-defined zero when all values are identical. And it isn’t the range, which is about the difference between max and min; that difference is also zero in this case, but the standard deviation specifically captures typical distance from the mean, which is zero here.

The standard deviation measures how far scores tend to be from the average. When every score is the same, there is no variability at all, so all deviations from the mean are zero. Squaring those zeros and averaging them still gives zero, which means the standard deviation is zero. This holds for both population and sample calculations.

It’s not equal to the mean, since it reflects spread, not central value. It isn’t undefined, because the usual formulas produce a well-defined zero when all values are identical. And it isn’t the range, which is about the difference between max and min; that difference is also zero in this case, but the standard deviation specifically captures typical distance from the mean, which is zero here.