Central Limit Theorem

A little formally, probability distribution of the sum (or average) of i.i.d. variables with finite variance approaches a normal distribution.

informally, say we have a probability distribution of integers. now, if we randomly take N numbers and then calculate then mean of those taken numbers (say it "sample mean"). Then the distribution of "sample mean" will be a normal distribution (with same mean µ and standard deviation σ/√N ).

As N increases, the standard deviation will decrease and the distribution will be more like a true normal distribution.

A nice explanation from Khan Academy.