# Motivating the Central Limit Theorem

## Main Question or Discussion Point

Hi, All:
I will be teaching intro Stats next semester, and I always have trouble making the CLT seem relevant/meaningful to students without much math nor probability background, whose eyes glaze at the mention of the distribution of the sampling mean being normal, no matter (given random selection, etc.) the distribution of the original population. Just wondering if someone has had success in making the CLT appear interesting and explaining its usefulness at this level.
Thanks.

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Some concrete examples would help a great deal, e.g. in dice games the distribution of 3d6 is close to a bell curve.

Some concrete examples would help a great deal, e.g. in dice games the distribution of 3d6 is close to a bell curve.
What is 3d6?

chiro
Hi, All:
I will be teaching intro Stats next semester, and I always have trouble making the CLT seem relevant/meaningful to students without much math nor probability background, whose eyes glaze at the mention of the distribution of the sampling mean being normal, no matter (given random selection, etc.) the distribution of the original population. Just wondering if someone has had success in making the CLT appear interesting and explaining its usefulness at this level.
Thanks.
I know this is probably too complicated, but when we learned the CLT and other asymptotic results, we used statistical software to simulate a variety of random variables and then got the software to produce some graphical properties of the distributions that showed how these converged to a particular distribution like say the normal.

Maybe there is a program you could use for this? Even if you created the document yourself so that you didn't require the students to generate the data, the students could be told to graph the distributions and see for themselves that the asymptotic results seems to be true.

Stephen Tashi
Judging by posts on this forum, many students don't understand the difference to be exptected between the shape of a histogram of many samples drawn from some non-normal probability distribution versus the histogram of the mean of many samples of some fixed size. A primitive intuition that is helpful is the thought that larger sample sizes make it more likely that extremes will "cancel out in the average". We can also consider the mistaken intuition that if we made sample sizes large enough the their means would always be the same because of this cancellation. The central limit theorem can be viewed as putting a limit of how effective that cancellation can be. (Admittedly this is a pun on "limit", but it's a useful one for purposes of teaching.)

An interesting demonstration (using computer software, of course) would be to have 10,000 samples drawn from a ramped shape distribution and then summarize this data 3 ways: 1) historgram the individual samples. 2) Group the samples in batches of 10 and histogram their means 3) Group the samples in batches of 100 and histogram their means.

AlephZero