Check for significant difference between numbers of sightings per cardinal direction I have data in a form like this:
quantity direction
10 n
5 e
6 ne
12 n
20 nw
5 s
8 n
1 sw
3 se
2 ne
6 nw
8 n
2 se
3 e
4 w
9 nw

The information behind these entries is that there were 10 animals (birds) moving north, 5 east, 6 northeast, 12 north and so on.
When I sum up these values I get:
38 n
8 ne
8 e
5 se
5 s
1 sw
4 w
35 nw

It seems that the directions north and northwest are the most favoured directions. I know I could run an ANOVA or Kruskal-Wallis H Test on my data (what I already did on the original much bigger dataset) to analyse if there is a difference between the flock sizes per direction. But is there a way to statistically prove that north and northwest are the directions which attract most of the birds whereas southwest and west are rather unpopular?
 A: First off, methods such as ANOVA and Kruskal-Wallis pay no attention to the circular nature of data such as yours. It's not clear how you imagine applying either, but if you intend to regard direction as a categorical predictor, you would be ignoring any structure to the measurement scale other than the directions being different. 
More generally, note that statistical methods essentially do not provide proofs; at best, they provide indications, point to conclusions, or aid decisions.
Your problem falls squarely within circular statistics, itself part of the statistics of directions. Problems to do with animal movements have historically helped stimulate this field. 
An appropriate test of a null hypothesis of uniformity of direction as compared with an alternative hypothesis of unimodality, i.e. whether there is a single preferred direction, is the Rayleigh test, which for your data yields a $P$-value of about 0.003, i.e. the idea of a preferred direction is well supported. 
More interesting and more important is what that preferred direction is, and my calculations yield a vector mean of 348$^\circ$, 12$^\circ$ W of N. 
The vector strength is 58%. That is a measure of variability with a minimum of 0 and a maximum of 1 or 100% if all directions agree. It is more commonly known as the mean resultant length; it can be considered as a measure of consistency, an excellent term occasionally used in this sense in circular statistics. 
Here is a graph produced in Stata showing the results of calculations with user-written programs (my own). Circular statistics are also well supported in R. 

I recommend a circular histogram like this, with bars proportional to frequency sitting on the sides of a regular polygon, over the much more common rose diagrams, which often are dimensionally ambiguous (is it sector radius or sector area that represents frequency?) and rarely avoid an ugly convergence of sector boundaries at their centre. 
I'd also suggest, although this is a little controversial, that for data like yours a standard histogram works well, so long as you rotate the scale so that mean or modal directions are roughly in the middle. 

https://en.wikipedia.org/wiki/Directional_statistics gives an introduction and some excellent references. The books recommended there by E. Batschelet and N.I. Fisher are scientist-friendly. 
