The answer depends on greater standard deviation that WHAT!
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It is a measure of the spread of the distribution. The greater the standard deviation the more variety there is in the observations.
It is the value that is one standard deviation greater than the mean of a Normal (Gaussian) distribution.
The standard deviation must be greater than or equal to zero.
In general, a mean can be greater or less than the standard deviation.
Yes. Since the standard deviation is defined as the square root of the variance, it can be said that the higher the standard deviation, the higher the variance.