Thank you! Fact is, this is not answering my problem: I don't simply need to smooth some data, not caring if my curve will touch or not the previous points, I need a curve that, if my series has a maximum, pass over it and then comes back down (you can see what I'm talking about in the graph I drew), a curve created with a smoothing technique would pass under the maximum, keeping its distance.

Vysochanskij–Petunin inequality is more efficent than Chebychev's, so it doesn't need a greater sample at all, but it has some use constraints: first, you have to have a continuous distribution, than, it has to be unimodal (no local modes are allowed). It may seem strange to drop normality assumption to adopt another one, but if your data is not discrete, sample mean should eliminate local modes even with very small samples. Fact is that mean has much of a bell distribution and, also if it can be skewed or have fat tails, it quickly comes to only have one mode.

I made a fast google search and I found out that binomial distribution is actually often used to explain different sample size need for skewed data, but I didn't find, and I guess there is no accepted "rate of convergence in terms of the skewdness".

higher entropy. I see you don't understand how does it all work, so I suggest you to read some more complete explanations, like: stackoverflow.com/questions/1859554/…, or math.unipd.it/~aiolli/corsi/0708/IR/Lez12.pdf (first pages only)

no, entropy of a node wtih 2 yes and 2 no is higher than one with 3 yes and 5 no, so there's an information loss. however, this is averaged with the information gain from the other node, which has a lower entropy, and the overall information gain is positive

with those bounds, variance can't be greater than 0.25, much better than 1, isn't it?

wikipedia docet

you choose the variable for performing the split for the information gain that you will get choosing it, and that's calculeted on nodes created by the split, so, yes.

so, you just want to convert a nominal variable to continous? or something more?