Central Limit Theorem

The Central Limit Theorem states that if a sum of independent and identically-distributed random variables has a finite variance, then it will be approximately normally distributed and the sampling distribution will have the same mean as the population, but the variance divided by the sample size.

In short the Central Limit Theorem states that the sum of a number of random variables with finite variances will tend to a normal distribution as the number of variables grows.

The smaller variance is intuitivally understandable if one just imagines that one size variation in the sample can compensate another.