Coefficient of variation of circular data The definition of coefficient of variation is as follows:
coefficient of variation= standard deviation/mean.
I am using circular (directional) statistics to find the mean and standard deviation. My question is: can I use the same definition of the coefficient of variation (given above) for circular data as well? Thanks in advance
 A: This answer argues that there is no need for any version of coefficient of variation for circular data. Worse, attempts to define or calculate one lead to absurdity.
For a self-contained introduction to the topic setting out some notation and definitions this Wikipedia article will do fine.
Circular data take values on an outcome space that is a circle. Direction as a bearing relative to North or some other standard direction is a common example. Let's suppose for concreteness that the units of measurement are degrees, ranging from $0$ to $360^\circ$, noting that the key point is that $0$ and $360$ degrees are identical directions. Nothing depends on a choice of units: radians or time of day or time of year could make as much or more sense for particular applications.
Let's back up and consider that any kind of coefficient of variation is the ratio of a standard deviation, or at least some measure of scale or variability, and a mean, or at least some measure of level or location.
The standard definition of a mean for circular data is the vector mean. A standard mean is very often a poor choice, as for example if you have directions that are $1$ and $359^\circ$ feeding them to any ordinary mean routine will return $180^\circ$. The vector mean is the arctangent of the sum of sines of directions divided by the sum of cosines, or more plainly the resultant from adding directions as vectors geometrically, namely end to end.  The vector mean of $1$ and $359^\circ$ is $0$ or $360^\circ$, which as a diagram will suggest is a natural solution. More generally, the vector mean is well defined in all but a few pathological cases (such as data that are two opposite directions, which cancel).
Variability of directions on a circle is most simply measured as mean resultant length. (That name is the most popular among several in use; in my view the rarely used name consistency is much more evocative.) The mean resultant length is at its largest when all directions are identical (hence the alternative name consistency) and at its smallest when directions are equally common in  opposite directions (a circular uniform distribution is one but not the only possibility). Hence mean resultant length is an inverse measure of variability, and some have defined circular variance as its complement (in 1 or in 100%). This is likely to seem confusing at first sight, as either measure is reported  on a scale from 0 to 1 (or 0 to 100%). But any convention about using degrees, radians or other units for the original data does not bite, as those units wash out of the definition and calculation. Further, a circular standard deviation has been defined but, again confusingly at first sight, not as the square root of the variance. For details, see for example the article cited above.
To the point: Dividing any measure of circular variability by any mean is neither needed nor even justifiable. The vector mean could be zero, i.e. it could coincide with the reference direction, whether say North, or South, or midnight, or the start of a year. If the absurdity of dividing by zero is avoided by rotation, then one absurdity becomes another as now the result of dividing any measure of variability by any vector mean is utterly dependent on a convention about the reference direction.
Coefficients of variation have often been oversold in statistical applications, but their merit when they are well defined and useful is as measures of relative variability of counts or measurements of variables with positive mean and an unequivocal zero point. See for example this thread on CV. There is no analogue of that set-up in circular statistics, which concerns a quite different outcome space.
A: This type of coefficient of variation is generally only useful if the mean is not 'arbitrary', that is, the data means something very different if the mean is artificially raised. An example would be count data, but there are many practical situations where the coefficient of variation could be sensible with continuous data as well.
For circular data, however, we rarely have this situation; in most cases we expect to be able to rotate the data without changing its meaning too much. That is, the point where we put $0^\circ$ is arbitrary. So, in most cases, a coefficient of variation is not useful; on the flip side, the standard deviation is more directly interpretable, as it has a 'maximum' if the data is circular uniform.
An exception could be 'accuracy' style data, where $0^\circ$ is 'hitting the target'. In these cases, the $0^\circ$ point is not arbitrary, and the coefficient of variation can be useful. The exception to this exception is if the mean is very close to $0^\circ$ most of the time; this would make the coefficient of variation unstable due to the division with a small number.
A: Given that you estimate the mean value and the standard deviation taking into account the circular statistics I suppose that you can estimate the CV with the same formula sd/mean. For example in the case of wind direction the well-known formula it works fine.
