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?
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).
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.
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