Any tips in explaining the central limit theorem in statistics?

Question

Many of my students don't have a mathematical background, and are not comfortable with concepts such as limits, random variables and distributions. Is there any intuitive way of explaining why the means of a random variable will form a normal distribution around the true mean? The goal here is not to produce a rigorous proof, as they are not mathematically oriented, but to demonstrate that the theorem works and teach them the "logic" behind it.

I thought about using computer programming and simulations to show that it works, but I feel that sampling from a simulated data is still too abstract to be intuitive.

Answer 1

This is a preliminary, not about a distribution of means: How about having every student measure your desk? Ask them to keep their measurements secret, so they don't influence each other (an interesting issue in itself). Then use their answers as a data set. If you have enough students, you should get a normal distribution, whose mean is the proper answer. (This is helpful for seeing why statistics is so important to science, and is much like my second, more relevant suggestion, therefore preparing them for it.)

Now do something similar with means. What is a mean that you and your students can identify with, something that they could get somewhat random samples for? Probably the hours that people watch TV doesn't clump too much. If they each ask 15 people (awfully small sample size, but not constructive to ask for more), and give the mean of their own sample, you can show the distribution of these means. They should form a normal distribution if your class is big enough. If your population isn't unusual in its TV watching habits, the mean of this distribution should be the true mean.

Answer 2

In my experience, there are a few key things that students misunderstand around the central limit theorem.

Firstly, many students don't actually have a concept of distribution in the first place. They need the idea that anything you could record about an object/person/group has various possibilities for what the outcome could be, some of which might be more likely than others. The description of what the possible outcomes are and how likely they all are is called the distribution. This description is sometimes a table of probabilities, sometimes a formula, sometimes a graph; and sometimes, the pattern of probabilities is so common and so useful it has a name, and just the name is enough to tell a statistician everything they need to know. The normal distribution is one of these.

Secondly, students don't have any distributions in their heads that aren't normal! When they think of a measurement with a mean, the exercises they've done so far have required them to draw a normal distribution anyway. So it's hardly surprising or useful to them that the mean of the means is normal because they think, well isn't everything anyway? Therefore, it's probably useful to show some distributions of things that aren't normal so that they appreciate what it might mean for it to be normal.

Thirdly, and most importantly, many students don't have a concept that the sample mean has a distribution at all! To them the sample mean is one value they calculate so how could it have a distribution? It is important to stress that anything that can be recorded or calculated about an object or group of objects has a distribution. That is, there is a description of all the outcomes that could possibly be and how likely they are. This point is almost never made explicitly about groups of objects, and I feel it's a big gap in most stats teaching.

As to how to deal with these things, well, here is what I would do. I have only actually tried some of these things as presentations rather than as class activities, so if you do try them I'd love to hear how they worked out!

First, I would list several things you could record/calculate about a group of objects and show graphs of the distributions of these things: the mean height of a group of 6 people, the standard deviation of the IQ scores of 10 people, the median of the prices of 100 houses, the maximum of the potassium level in 5 bananas, etc. The point you are trying to make is that any measurement you could calculate has a distribution, and this distribution might or might not be something recognisable.

If at all possible, I would get everyone to do their own little sample. It could be as mundane as getting everyone to roll a die 5 times and record the result, or as interesting as getting them to measure the length of 5 snake lollies. If you can't get them to do it themselves, you could always provide everyone with their own 5 measurements, but I think it would help if it was an everyday thing they could measure themselves if they wanted to.

Then get everyone to calculate various things from their sample and record the distributions of them on a big graph on the board. I'd recommend doing a couple of crazy calculations, like "the smallest, plus the biggest, minus two of the middle one", just to really reinfoce that any calculation has a distribution. This should give them a real feel for how different calculations produce different distributions. It should be clear that the mean has a particularly nice distribution.

Then you could do it with a bigger sample size, maybe this time with just the mean. After having done this twice, it's probably ok to proceed by just showing them the results that you would have gotten rather than getting them to do it.

Now you can show how the distribution of the possible means is nicely clumped in the centre and more so as the sample size increases. And you can say that it's not just clumped in the centre, there is a specific well-known distribution that describes this shape -- the normal distribution.

A final comment: the fact that the distribution of sample means is approximately normal is actually not the point, of course. The point is that it's really useful to know that it will come out this way because then you can use everything that mathematicians know about normal distributions to say something useful. Doing all the preceding is probably a waste of time without some discussion of why you care so much about it!

Answer 3

I personally find a Galton board more compelling than a computer simulation. There are lots of YouTube demonstrations; this is one of the better ones I found after a very brief search.

Answer 4

One thing that I think can be made intuitive is that averaging several samples of a distribution gives a new distribution that is 'nicer'. You can explain that taking the mean averages things out, and removes any weird assymetries, 'lumps' or 'corners' in the distribution (you can illustrate this by plotting the distribution of the mean of a random variable with increasing sample size.

Thus, the means of all random variables will converge to the distribution that is most smooth and most symmetrical. At this point, you claim that the normal distribution is the smoothest (which is one interpretation of the fact that it is fixed by fourier transforms).

This discussion ignores the fact that one must divide by a factor to keep the distributions normalized. One could just pick a distribution and show pictures of the mean distributions for a few values of $n$, and show how they look more like each other after rescaling by an appropriate factr.

Answer 5

Perhaps start with an uniform distribution between $-1$ and $1$, and show what the distribution of the sum (or average) of 2, 3, ... IID variables looks like? Not too hard to do analytically, easy to do by simulation (have the students write a small program generating the values, and use e.g. a spreadsheet or a plotting program to show histograms). To get a "feel" requires many values, and getting them experimentally is hard.

Answer 6

Is there any intuitive way of explaining why the means of a random variable will form a normal distribution around the true mean?

One can certainly motivate a fact that somewhat similar to this (at least under certain conditions), but that's not what the CLT says.

The actual CLT doesn't really say anything about what happens in samples of given finite size. Speaking hypothetically, it would be perfectly consistent with the CLT if sample means at no sample size ever observed - even into the trillions and higher - ever saw anything close to a normal distribution.

However, it is the case that (under certain conditions), standardized sample means at finite sample sizes do get closer to being standard normal - multiply the sample size by 4, say, and the distance between the CDF of the distribution of the standardized sample mean a standard normal will tend to halve.

It's a fact - and a very important fact - but the CLT itself does not establish that what we observe will necessarily happen. Other theorems come much nearer to explaining that we will observe it (the Berry-Esséen theorem, for example).

[The question of whether you should avoid attributing it to the CLT is a different question -- given many texts (etc) do it, it may not be worth bucking the trend over a relatively esoteric point that doesn't really alter the property you want to describe.]

I thought about using computer programming and simulations to show that it works, but I feel that sampling from a simulated data is still too abstract to be intuitive.

You can get somewhere with rolling dice (means of 1,4,16 rolls, say, and using results from the whole class to get distributions of averages), but it takes a while (keeping in mind it's really cdfs that are converging; trying to demonstrate convergence of a discrete pf to a continuous distribution will get you into trouble).

Outside of that, computer simulation can be very useful, particularly if they're able to do it themselves.

Answer 7

It should be possible to convey the idea of CLT graphically.

Represent the outcome of a dice roll by a histogram, six segment of height 1/6 standing over each number from 1 to 6. Then, represent the outcome of the sum of two dices rolls the same way, and more importantly show how one goes from the first histogram to the second. For example, represent first independently the two rolls, one above the other, and consider how each outcome of the second dice affects the sum: you could draw six histograms, one for each outcome of the second dice, with segments of height 1/36 placed at 2 to 7 for the first one, 3 to 8 for the second, etc. Then sum them into the histogram representing the sum of the two rolls. At this stage, you illustrated how sum of random variables have a distribution obtained by convolution (you don't have to say the big words).

Then represent in a sequence what happens when we draw more dices, and scale down the horizontal axis and scale up the vertical axis so that the picture fits in the same box; tell the students about the scaling. At this stage, they will see the normal distribution emerge.

Then do the same with another starting distribution, say one that looks quite random. Again, the normal distribution will appear, and this will show how universal the normal distribution is.

Then represent two normal distributions and represent their convolution. This will show the normal distribution as a fixed point (up to scaling and translation) of the operation of "taking the sum of random variables". Then what has to be accepted is that it is an attracting fixed point, but at this stage you give a pretty decent understanding of what is going on.

In a nutshell, looking at sum of random variable one performs a certain operation on their distribution (hardly surprising) and rescaling to only retain the "shape" of the distribution, we get an operation that has an attractive fixed point, something not obvious but a plausible outcome in such a situation. The normal distribution is important because it happens to be this fixed point. The fact that the starting distribution does not matter seems less magical once one has understood this dynamical point of view; it is pretty much the same phenomenon as in simpler systems : e.g. if we start from any number, divide it by 2 and then add 1, and perform this operation over and over, whatever the starting number we end up closer and closer to 2. (You can even start with this before showing how it goes for the CLT).

Last, if you want to warn them about how important it is to only use a theorem when its hypotheses are verified, you can do represent the convolution of a Cauchy distribution with itself, to show that it gives another fixed point! This will show that the normal distribution is not universal when the second moment is not finite.

Answer 8

In my community-college statistics course I motivate this by taking a minimal discrete population {1, 2, 3} and listing the sampling distributions with replacement for sample sizes 1, 2, and 3 (which are respectively: uniform, triangular, and bell-shaped). Specifically: I do n = 1, students do n = 2, then I present a slide for n = 3 and have students name the shape. I think this concrete example is about the best thing we can do to lend intuition about what we're discussing. I got the idea from this Wikipedia article:

https://en.wikipedia.org/wiki/Illustration_of_the_central_limit_theorem#Illustration_of_the_discrete_case

Answer 9

I would use a quincunx (also known as Galton board). I realize that has been answered before, but here I will give some images of how a quincunx could look like. First a photo (by myself) of a very large quincunx in the science museum http://www.universum.unam.mx/ in the large UNAM camp in south Mexico City:

This is very large, about three meters long if I remember right, and works by being tipped like a childrens seesaw. Below another image showing better the principle: