I'm teaching both at the same time to different classes in high school, so I just wondered about this.
Added by OP on 16.May.2011 (Beijing time)
I mean Statistics only, without Probability. In other words, Descriptive Statistics only. This rules out Buffon’s Needle Problem.
The occurrence of π is counted only as a connection to geometry. By Trigonometry is meant explicit, non-gratuitous, occurrence of the sine, cosine, tangent, or their reciprocals.
Yes, the Law of Cosines fits the bill, but it is on the surface: everyone knows about it. It would be a hugely interesting meta theorem that this were the “deepest” connection between Trigonometry and Descriptive Statistics. My suspicion/hope is that there are deeper connections, somewhat along the line of the surprising use of trig in solving the cubic in closed form, or the use of trigonometric substitutions in evaluating certain integrals. The comment below about the arcsine transformation at first blush seems to be something along this line, but when you follow the link you see that someone is bringing it up only to say how bad it is.
So, I hope the intent of my question is now much clearer.