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)
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I mean Statistics only, without Probability. In other words, Descriptive Statistics only. This rules out Buffon’s Needle Problem.
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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.
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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.
Best Answer
No doubt it's the Law of Cosines. The correlation between two data sets follows the generalized $n$-dimensional Law of Cosines.
EDIT: Maybe I'll make this a little more explicit. Take two data sets $A = (5,7,2...)$ and $B = (12, 4, 9...)$ and ask if they are correlated. One way is to treat them as vectors and look at the data set $C = A+B = (17, 11, 11...)$ where the sum of the data sets (vectors) is taken pointwise. Okay...it's not the length of the vectors that works like the Law of Cosines, but the standard deviation. If the two data sets are randomly correlated then you should expect the standard deviations to add like the Law of Pythagoras, so that if $\text{StDev}(A) = 3$ and $\text{StDev}(B)=4$ then $\text{StDev}(C)$ should equal $5$. For $100\%$ correlations, the standard deviation of $C$ would have to be $7$ (or $1$ for negative correlation). It's the Law of Cosines where the correlation is the cosine of the angle between two vectors of length $3$ and $4$.