Sometimes in english texts I see this type of abbrevation of numbers .58, .278. A number without any number before the dott. Have you seen it before? What does it mean? Does it mean 'zero-dott-something' and they simply leave the zero out?

Yes. I think that's because 0.56 is commonly read "point fifty six".

This looks like a batting average.

They are decimal numbers which represent the fractions or numbers between the whole numbers. I'm sure you've learned this in math or dealing with money. In America you can freely right .xx or 0.xx except for, I believe in terms of money where you almost always write $0.xx.

Thanks guys!