stats noob here.
I have a (circular) dataset with values in [0, 2pi]. I need some kind of a measure of how disperse or diverse the dataset is. I have looked a bit into non circular (regular) data, and I happened to come across coefficient of variation as a non-dimensional measure of dispersion in a dataset.
My question is: can I modify the coefficient of variation in my case as (circular stddev) / (circular mean) and use it as a dispersion measure?
Given the finite support of the domain, and the fact that the most dispersed distribution in this case is certainly the uniform distribution $U$ (with $u(x) = 1/2\pi$), you could measure the dispersion via the Kullback-Leibler divergence from uniformity, which in this case is closely related to the entropy of the distribution:
$$ D_{KL}(P \mid \mid U) = - \int_0^{2\pi} p(x) \log \left( \frac{p(x)}{1/2\pi} \right) \, dx = \log \left(\frac{1}{2 \pi} \right) - \int_0^{2\pi} p(x) \log p(x) \, dx $$
One reason to suppose that this could be preferable to standard deviation (or the c.v., which does have a problematic zero divisor case) is that this is invariant under rotations of the plane in which $x$ measures angles -- the value $x = 0 \, \textrm{rad}$ is usually an arbitrary marker point in cases where circular distributions would be use, and any other point on the circle could have been chosen as $x' = 0 \, \textrm{rad}$. Because the definition only involves a total integral and the values of $p(x)$, the result is the same no matter what reference direction is chosen as 0 radians.
The standard deviation of a distribution on on the real line is useful exactly because is invariant to linear translations of the axis, but fails to respect the underlying symmetry for the circular domain (think, for instance, if the probability spiked at a value just greater than $0$ and another value at just less than $2\pi$, the standard deviation would be large, but $0$ and $2\pi$ are close in angular space -- with a different choice of reference direction these points could have numerically similar angles, and the distribution would have small standard deviation.
Short answer is No. A coefficient of variation such as you define it is not just useless, it is meaningless.
For example, two datasets that are 350, 0, 10 degrees and 170, 180, 190 degrees have (I suggest) equal dispersion by any standard BUT dividing any measure of dispersion by the mean makes no sense, even if it is a vector mean. In other words, the mean can easily be zero.
(In this example, and rarely, the vector mean and the ordinary mean coincide for each dataset.)
More generally, the position of the mean depends on a convention about what is zero direction. In geography and Earth sciences, North as a bearing is usually zero, but there could be excellent grounds for using another direction as zero. The same goes for time of day, time of year and any other circular outcome space.
But a positive answer is that several measures of dispersion are defined for circular data. The range can often be useful, defined as the complement of the largest gap on the circle. So, the two toy examples above both have range 20 degrees: that is obvious enough for 170, 180, 190 and obvious when you think about it for the other example. The mean resultant length is more nearly standard. (It has many other names, including vector strength and consistency.)
There is an entire literature on circular statistics with several dedicated monographs, for all that many statistical people never have cause to know about it.
I don't think this directly answers your question but you seem to be looking for a way to measure dispersion in circular data. As others have pointed out, there is a literature on circular statistics and standard solutions to this problem. The basic summary statistic for circular (or otherwise cyclic) data is the mean vector.
Assuming your data is not weighted, you interpret every angle $\theta_i$ as a unit vector.
$$ \bar{v}_i = [\text{cos}(\theta_i), \text{sin}(\theta_i)]$$
Your mean is then given by:
$$ \bar{r} = \frac{1}{n}\sum_{n} \bar{v}_i $$
The angle of $\bar{r}$ ($\theta_\bar{r} = \text{atan2}(y_\bar{r}, x_\bar{r}$) gives you the mean angle of the data. The magnitude of $\bar{r}$ ($|\bar{r}| = \sqrt{x^2 + y^2}$) ranges from 0 (data is completely dispersed, uniform) to 1 (all data points are identical). Circular standard deviation and angular deviation can both be computed from $|\bar{r}|$.
There are also a number of tests on dispersion, the most basic of which is the Rayleigh test (is the data sufficiently concentrated in a particular direction or not).
If you're a stats noob (like me) I would recommend something like Batschelet (1981), Circular Statistics in Biology. It gives a very easy introduction which explains why circular and linear measures should not generally be mixed and then provides a 'cookbook' of different tests and the assumptions which must be satisfied in order for them to be valid.
