# What intuitive explanation is there for the central limit theorem?

In several different contexts we invoke the central limit theorem to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I understand the technical details as to why the theorem is true but it just now occurred to me that I do not really understand the intuition behind the central limit theorem.

So, what is the intuition behind the central limit theorem?

Layman explanations would be ideal. If some technical detail is needed please assume that I understand the concepts of a pdf, cdf, random variable etc but have no knowledge of convergence concepts, characteristic functions or anything to do with measure theory.

• Good question, although my immediate reaction, backed up by my limited experience of teaching this, is that the CLT isn't initially at all intuitive to most people. If anything, it's counter-intuitive! – onestop Oct 19 '10 at 2:39
• @onestop AMEN! staring at the binomial distribution with p = 1/2 as n increases does show the CLT is lurking - but the intuition for it has always escaped me. – ronaf Oct 19 '10 at 3:18
• Similar question with some nice ideas: stats.stackexchange.com/questions/643/… – user88 Oct 19 '10 at 6:42
• Not an explanation but this simulation can be helpful understanding it. – David Lane May 12 '17 at 18:12