# Demonstration of central limit theorem

I teach basic (very) statistics to prisoners in a medium/high security prison and would like to demonstrate the Central Limit Theorem. The classroom has no resources beyond a white board. I can only bring in paper and writing implements. Any suggestions on a simple demonstration?

• Several of the answers to stats.stackexchange.com/questions/3734/… address this (but they are not comprehensive). What other props might be available? Coins, perhaps?
– whuber
Nov 11, 2014 at 23:01
• If you can bring in a piece of paper tape measure (where I live, IKEA gives these away, but you can print one easily enough onto paper), you can measure things (if I recall right, Student himself looked at averages of finger lengths over small groups when developing the t-test). If the students all carry something you can measure or count (maybe cigarettes?) you can look at distributions of averages of small groups compared to distribution of individuals. But these will work better if your class size is large; in small groups they may not do enough to show what you mean. Nov 12, 2014 at 0:58
• @Glen_b Yes - moreover they were the finger lengths of convicted criminals! The data set Gosset a.k.a. Student used is actually available for R. stat.ethz.ch/R-manual/R-patched/library/datasets/html/… Nov 12, 2014 at 1:29
• Looks like it was averages of samples of size 4; both height and left middle finger length were considered, but he had 750 averages of that size to play with, which you won't have. If you have only about 20 students you probably won't see enough even with small samples (you need enough averages to see the effect on shape). But if you have a hundred, you might see something. Nov 12, 2014 at 1:41

I assume that by "demonstration" you mean, "showing what it's about", not a mathematical demonstration.

I would draw a Galton board on the black-board, and simulate what happens as you drop balls, making a random choice each time. You can even ask the students to pick "left or right" randomly a few times, to make it clear that the process is random and you're not deliberately choosing the path (though you should probably do so, in order to get better convergence).

You could also ask all the students their height, and plot a histogram. Why does it look like a bell curve? It's a contribution of many random effects.

• This seems like a good idea. If you have a coin to flip, you could use that to make your choice of left or right. If they don't get coins, but are allowed a pen, you could spin that for l/r. Nov 12, 2014 at 0:59

If you are in US, this url has statistics on US prisons

Perhaps you could explore it to see whether some manifestation of the CLT emerges in there.

A proposal,

tell the people to choose papers (or cards) from a pack of cards where each card has a number from -2 to 2.

Also have 5 boxes (A, B, C, D, E).

At first time tell them to choose only one card each.

Then tell them to place their cards into one of the boxes.

When everyone (including you if you want) has placed her card(s) in a box, count the sum of each box and draw a histogram, this wil be very crude.

Then tell the people to choose more than one card (make sure there are enough cards for the experiment) and place their cards on a box (or even different boxes).

Repeat the counting process, draw a new histogram

Repeat once again with people having even more cards each, repeat the counting process, draw histogram.

One can repeat this process as many times as one wants.

Observe the histograms and their shapes.

Can you bring in dice? (or maybe they already have access to dice). Then have them throw the dice (many times, they have the time ...), draw histograms, calculating mean number of eyes, drawing histograms of that for different $n$, so on ...

With many dice you can even build the histogram directly with the dice. We got this in class: