The Central Limit Theorem

This is an interactive demo meant to illustrate the Central Limit Theorem from Probability, or at least the general idea.

Click one of the buttons below the graphs to roll a single die. The result will be recorded in the bar graph.

Now try rolling a single die multiple times. You can either click the "Roll 1 Die" button over and over, or do a bunch of iterations with the other buttons. The graph should update with the total counts. You can always refresh the page to reset the data.

Some questions to consider after rolling a bunch of times:

• What sort of shape does the bar graph take?
• Why does it have that shape?
• What do you expect the value of a single roll to be? (Can you think of any reasonable answer??)

Instead of just rolling one die, let's roll two at a time. You can use the graph and buttons below to simulate this. The graph will record now the average of the dice roll. So if you roll 3 & 6, the bar for 4.5 will increase.

Roll the dice pairs a bunch of times, then reflect on the following:

• What shape does the graph take now? Why is that? (Hint: What happens when extreme values are rolled on opposite sides?)
• What would you expect the value of the average to be most often? Does it match the picture?

Now we have a pack of 25 dice! But before you get rolling them and plotting the average of the roll, let's think about the questions first.

• What sort of shape should the graph take after rolling a bunch of 25-packs?
• Come up with some really good reasoning for your prediction!
• What should the average of the roll be around, typically?

After rolling, check your answers from before. Then think a bit more about these:

• How often do you expect the average of the roll to be 1? Can you quantify this at all?
• Since things are random, it's certainly possible to get an average of 1. How do the chances in this case compare with when we were rolling two dice?
• How do the chances compare with rolling an average of 2.5 in this case compared to the chances of rolling an average of 2.5 in the previous case? Do you have an intuitive explanation for that? (Hint: Think about opposite extremes again.)

OK, here's the real test of what we've seen so far. I've just handed you a pack of 400 dice. First, think out analogues of the above questions (expected shape? average?), then roll them.

Let's refine our intuition:

• First: what's the shape? Can you explain it intuitively?
• What average of the roll do you expect? How do the chances of rolling 2.5 compare with the previous cases?
• If you're feeling ambitious, try to quantify your thoughts from the last question. How does increasing the number of dice rolled at once change how spread out the averages are? The x-axis has the same scale in all of the graphs, so you can go back and compare.

After you've thought about the above, the main intuition of the Central Limit Theorem should be clearer. The ideas above apply quite generally when measuring the average of some random sample. The sampling distributions drawn above get narrower as the sample size increases because the larger the sample, the more opportunities for opposite extreme values to balance each other towards the center there are. So in every case the distribution of the sample means will be centered around the same "average," but the variance/standard deviation decreases. This leads to the very practical conclusion that as the sample size grows large, we can feel more confident that our sample mean reflects the population mean, because it almost certainly lies in narrow spike centered around the population mean. The Central Limit Theorem quantifies this intuition to give a precise prediction of what the standard deviation of the sample distribution should be.