Intuition behind the Central Limit Theorem

Central Limit Theorem:

If X is the random variable of a population distribution having finite[1] population mean (u) & finite variance (σ²), & we do sampling from X with each sample size as n, then the distribution of sample means will follow a Gaussian / Normal distribution with mean same as u (population mean) & variance as σ²/n as n tends yo infinity. 

[1] - (So you may ask is it possible for a distribution to have infinite means & variance? The answer is,  Yes, there can be certain distributions for which mean & variance are either infinite or are not defined. s.a. Pareto Distribution. Though there are extensions of Central Limit Theorem applicable for such distributions as well)

Central limit theorem : The tendency of the sampling distribution to take on a normal shape as sample size rises.