thermodynamics - Is speed an intensive property?

I remember being taught in elementary physics that while it makes sense to add volumes, masses, or heat, it makes no sense to add temperatures.

As I wanted to use that to illustate some other issue, I checked it on wikipedia, and discovered the concept of intensive and extensive properties of materials and systems, which is apparently a century old, though I do not remember it being taught to me (it was not as old then :-).

The concept seems to be particularly useful in material and systems thermodynamics. But I was wondering how far it extends, and I was keeping in mind my original problem of determining when adding values in a given unit (not necessarily a physical one) made sense or did not.

I noticed the fact that the ratio of two extensive variables is intensive. So I started looking, a bit ramdomly I confess, at ratios, and the first that came to mind was speed. And I wondered whether adding speeds made sense.

We do that all the time, so it should. But then, my initial problem was not about adding two quantities, but a long list of them (it was a database question). While adding a few speeds (or velocities) makes sense when I move in the bus, or analyse the motion of the Moon in the solar system, I cannot imagine it would ever make sense to add a list of a hundred or a thousand speeds (though computing the average would make sense, as it would for temperature). But I makes perfect sense to sum the masses of a thousand objects.

So there is apparently something special, not quite right, about adding speeds. Of course, I know that slightly older results state that adding speeds is not done with simple addition, but I feel that my problem is elsewhere (though there may possibly be a connection).

I am aware that temperature in some materials is related to speed of motions inside it, so I am not too surprised. But speed in general seems to go beyond that (sorry for the vagueness). Also speed is not listed by Wikipedia as an intensive property.

So, my question is : why does it seem improper to add many speeds (or velocities)?

I guess this must also be true of other physical quantities, and I am wondering what is the right way to look at this, and understand it. Is there a more general notion than intensive and extensive?