[zeroonetwothree]

I don’t understand the picture with the shaded circle. Sure the area to the left is smaller, but it also is more likely to be chosen because in a Gaussian values closer to the mean are more likely. So the picture alone doesn’t prove anything.

[rssoconnor]

In the diagram the mean of the distribution is the center of the circle.

Of the set of samples a fixed distance d from the mean of the distribution, strictly less than half of them will be closer to the origin than the mean is, and strictly greater than half of them will be further from the the origin than the mean. This is true for all values of d > 0, so the result holds for all samples.