# How do you convey the beauty of the Central Limit Theorem to a non-statistician?

My father is a math enthusiast, but not interested in statistics much. It would be neat to try to illustrate some of the wonderful bits of statistics, and the CLT is a prime candidate. How would you convey the mathematical beauty and impact of the central limit theorem to a non-statistician?

What I loved most with CLT is the cases when it is not applicable -- this gives me a hope that the life is a bit more interesting that Gauss curve suggests. So show him the Cauchy distribution.

• what is the relation between the Cauchy distribution and the CLT or the failling of the CLT ? Jul 26, 2010 at 20:02
• @robin Consult en.wikipedia.org/wiki/Cauchy_distribution#Properties
– user88
Jul 26, 2010 at 20:12
• The CLT requires that the MGF's exist in a neighborhood of 0. The Cauchy distribution does not have that property. CLT Win. Cauchy doesn't even satisfy the weaker requirements of a stronger version of CLT where all that is required is that mean and variance exist. The Cauchy distribution shows that the mean is required to exist for the CLT to hold. It doesn't make the CLT fail. Jul 27, 2010 at 17:46
• @Baltimark You have misunderstood my post -- its obvious that Cachy is not covered by CLT because of CLT assumptions, otherwise it would be impossible to prove CLT. I have gave this example because people believe that CLT works for all distributions; probably "fail" is not a perfect word, but still I don't think it is a reason for downvote. Ok, I have even changed it to not applicable.
– user88
Jul 27, 2010 at 18:05
• I prefer your edit. The Cauchy distribution is definitely very cool. Jul 27, 2010 at 18:11

To fully appreciate the CLT, it should be seen.

Hence the notion of the bean machine and plenty of youtube videos for illustration.

• I thought it shows the binomial distribution; I don't think that its asymptotics have a direct link with CLT.
– user88
Jul 26, 2010 at 19:52
• bean machine by the author of the package animation... yihui.name/en/wp-content/uploads/2010/07/… Jul 26, 2010 at 19:55
• @mbq take a look at en.wikipedia.org/wiki/… Jul 26, 2010 at 20:01
• @robin I have wrote about it, what's the problem?
– user88
Jul 26, 2010 at 20:09
• @ShreevatsaR The point is that a good illustration should show the "core" of something, and (IMO at least) the "core" of CLT lies in the fact it melds many variables from different strange distributions into a gaussian, not that it is just a limit of a binomial distribution.
– user88
Nov 16, 2012 at 9:29

Often when mathematicians talk about probability they start with a known probability distribution then talk about the probability of events. The true value of the central limit theorem is that it allows us to use the normal distribution as an approximation in cases where we do not know the true distribution. You could ask your father a standard statistics question (but phrased as math) about what is the probability that the mean of a sample will be greater than a given value if the data comes from a distribution with mean mu and sd sigma, then see if he assumes a distribution (which you then say we don't know) or says that he needs to know the distribution. Then you can show that we can approximate the answer using the CLT in many cases.

For comparing math to stats, I like to use the mean value theorem of integration (which says that for an integral from a to b there exists a rectangle from a to b with the same area and the height of the rectangle is the average of the curve). The mathematician looks at this theorem and says "cool, I can use an integration to compute an average", while the statistician looks at the same theorem and says "cool, I can use an average to compute an integral".

I actually have cross stitched wall hangings in my office of the mean value theorem and the CLT (along with Bayes theorem).

• Hmmm. I would think that most mathematicians use the MVT to approximate an integral as a rectangle. Jun 9, 2011 at 22:54

I like to demonstrate sampling variation and essentially the Central Limit Theorem through an "in-class" exercise. Everybody in the class of say 100 students writes their age on a piece of paper. All pieces of paper are the same size and folded in the same fashion after I've calculated the average. This is the population and I calculate the average age. Then each student randomly selects 10 pieces of paper, writes down the ages and returns them to the bag. (S)he calculates the mean and passes the bag along to the next student. Eventually we have 100 samples of 10 students each estimating the population mean which we can describe through a histogram and some descriptive statistics.

We then repeat the demonstration this time using a set of 100 "opinions" that replicate some Yes/No question from recent polls e.g. If the (British General) election were called tomorrow would you consider voting for the British National Party. Students them sample 10 of these opinions.

At the end we've demonstrated sampling variation, the Central Limit Theorem, etc with both continuous and binary data.

Playing around with the following code, varying the value of M and choosing distributions other than the uniform can be a fun illustration.

N <- 10000
M <- 5
meanvals <- replicate(N, expr = {mean(runif(M,min=0, max=1))})
hist(meanvals, breaks=50, prob=TRUE)


In my experience the CLT is less useful than it appears. One never knows in the middle of a project whether n is large enough for the approximation to be adequate to the task. And for statistical testing, the CLT helps you protect the type I error but does little to keep the type II error at bay. For example, the t-test can have arbitrarily low power for large n when the data distribution is extremely skewed.

If you use Stata, you can use the -clt- command that creates graphs of sampling distributions, see