Can the standard deviation ever be negative?

I would suggest you to recall the formula for standard deviation.For instance, when we take the corrected sample standard deviation into account we know that;

##s = sqrt(1 /(N-1)sum_(i=1) ^N(x_i-bar x)^2 ##

Standard Deviation

As you can see, you need to take the square root of the above expression in order to find the standard deviation and we know that we cannot have a negative number inside the square root.

In addition, the ##N## stands for the size of the sample (group of people, animals etc.) which is a positive number and if you expand the second part of the expression ##sum_(i=1) ^N(x_i-bar x)^2## it is clear that you'll end up with having either zero or positive number as you have to square the differences from the mean.

Thus the inside of square root will be greater than or equal to zero and we will end up with having a non negative number for standard deviation so it doesn't make any sense to talk about the square root of a negative number.