This is the normalized histogram for bit strings of length 30. There are 290 bits in the triangle. The lines are at the mean and +- one standard deviation. A general rule of thumb for eyeballing the standard devation is to take the width of the density curve where reaches half it's max. In this case that's pretty close.

This is the log of the density above with a quadratic fit to the middle portion of the curve.
This is the density with an approximate fit by a StudentsT distribution.

Same as above but log scaled on the y-axis.