Data on a circle - testing if points are close to another given point I have a set of points that lie on a circle, and I know their angles.  Is there a way that I can test whether they are closer to another point x on the circle than expected by chance (assuming a uniform distribution on the circle)?
Can I calculate mean arc length between xand the data, and then calculate the same for each of many simulated data sets generated by the uniform distribution?
 A: The question as posed sounds like it's setting up a hypothesis test.
Distance in the case of circular data would (presumably) simply be angular distance which is the same as your suggested arc length when measured in radians on a unit circle. That is, for example based on the absolute difference in angle (which I think is your suggestion), or maybe something based on the squared difference in angle or some similar measure. 
The null is that the distribution of such distances should be that for a uniform distribution of points, while the alternative is that the distances to $x$ will be smaller (since you said you're checking for clustering around $x$). If you wanted to check for being further from $x$ you'd look for the alternative to be larger, not smaller.
Without loss of generality, you might as well just take $x$ at the centre of your uniform*; it won't change the null distribution.
So you'd be trying to work out the distribution of the mean absolute angle for a uniform distribution (or perhaps the mean of the squared angle for a uniform distribution).
And yes, you can certainly do that by simulation, but I suspect the algebra isn't very hard at all. (Edit: actually, I just figured it out now for your suggested statistic. It's pretty straightforward - it will be amenable to somewhat mechanical direct solution, but it's already one which is quite well-known. Practically speaking, in anything but quite small samples you'd probably just use a normal approximation unless your significance level was very low.)
[Actually, this would be a case on which one could use the trick in Fisher's method to base a statistic off the sum of the log-absolute-angle-distance. That has the (modest) advantage of having a standard table as the distribution of the test statistic.]
* if angles are on $(-\pi,\pi]$, you could take $x=0$, while if on $[0,2\pi)$, you could take $x=\pi$. 

If you measure the absolute angle from $x$, you get a uniform on $[0,\pi]$ (measured in radians). Any other angle-origin is easy to convert to this. The average of the absolute angles will therefore be a convolution of $n$ uniforms. For small $n$ this is easy to write explicitly (e.g. for $n=2$ the average is triangular, for $n=3$ it's a smooth-looking hill-shape composed of three quadratic segments). As $n$ increases it rapidly approaches normality except in the extreme tails.
You can base a test off that quite easily; the alternative (growing away) would correspond to larger-than-average absolute angle. Growing toward would be smaller-than-average absolute angle.
If you're happy using simulation, I see no problem with that, but if $n$ is bigger than about 15 or so you can probably manage quite well with the normal approximation.
A: If your data scatter only in a small region of the circle, you can take the tools of euclidean statistics. You would then use that a manifold --here the circle $S_1$-- is locally euclidean. This is your and Glen_b'a approach.
However, you exclude this case in your model: Your null hypothesis says that the data scatter around the whole circle. So you have to use directional statistics. The mathematical reason why you can not use euclidean statistics by mapping each point on the circle to it's distance from a particular $x\in S_1$, let's call it north pole, is that there is no homeomorphism $S_k \rightarrow \mathbb{R}^k$. That's why a distribution on a circle has no mean in the classical euclidean sense: What's the mean of 270° and 90°? Is it the south pole or the north pole?
Fortunately, there are already tests for your hypothesis. See e.g. Ajne, "A Simple Test for Uniformity of a Circular Distribution", Biometrika, 1968, 55 (2), p. 343. 
