Demonstration of the central limit theorem

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The Central Limit Theorem states that if random samples of size n are drawn again and again from a population with a finite mean, mu(y), and standard deviation, sigma(y), then when n is large, the distribution of the sample means will be approximately normal with mean equal to mu(y), and standard deviation equal to (sigma(y))/sqrt(n).

Let's examine the effects of the Central Limit Theorem with the following experiment. Suppose you toss a fair die 1000 times. You would expect to get about an equal number of 1's, 2's, and so on. Let's examine the distribution of 1000 tosses. This is shown in Graph 1.

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Now suppose you were to toss the die two times and take the average of the two tosses. You will repeat this experiment 1000 times also. Let's see what the distribution of the averages of two tosses looks like. This is shown in Graph 2.

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Did you notice that with only two tosses the distribution of the averages was already becoming mound-shape Suppose that you now toss the die three times and take the average of the three tosses. Again, you will repeat this experiment 1000 times. Let's see what effect this has on the distribution of the averages. This is shown in Graph 3.

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Again, the shape of the distribution is quite close to that of a normal distribution. Did you notice anything else that was happening to the distribution?

Let's toss the die five times and take the average. Again, you will repeat this experiment 1000 times. This is shown in Graph 4.

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Have you begun to notice any patterns in what is happening yet?

Let's continue to increase the number of tosses that we are averaging. This time you will toss the die 10 times and take the average of the 10 tosses. This is shown in Graph 5.

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By now you should see two phenomena as you increase the number of tosses. First, you should see that the shape of the distribution of averages is really beginning to take on the shape of a normal distribution. Second, you should see that as the number of tosses increases, the distribution becomes narrower and narrower. Let's continue increasing the number of tosses. This time you will toss the die 20 times. This is shown in Graph 6.

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You should by now be adequately convinced of the effects that increasing the sample size has on the distribution of sample averages. You will increase the sample size one more time to reinforce this thought. This time you will toss the die 30 times. This is shown in Graph 7.

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Let's review what you have seen.

You will draw the histograms for samples of size 2, 5, 10, 20, and 30 together in one plot to see the changes in the distribution.

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The Central Limit Theorem tells us what you should have seen, theoretically. Lets' compare this to what you actually did see:

Theoretical Results Observed Results ------------------- ---------------- Sample Standard Standard Size Mean Deviation Mean Deviation ------ ---- --------- ----- --------- 1 3.5 1.707825 3.453 1.7041 2 3.5 1.207615 3.527 1.2320 3 3.5 0.986013 3.546 0.9503 5 3.5 0.763763 3.481 0.7532 10 3.5 0.540062 3.506 0.5289 20 3.5 0.381879 3.510 0.3891 30 3.5 0.311805 3.507 0.3148