The Box Model Part 2 - Central Limit Theorem

Consider that we have a box model with one '1' ticket and three '0' tickets. Each sample consists of 25 draws from the box, and we represent the sample using the mean of the tickets.

For example, imagine that from one experiment, we drew 10 '1' tickets and 15 '0' tickets. The sample mean would then be:

$$\frac{10 \times 1 + 15 \times 0}{25} = \frac{10}{25} = 0.4$$

But this example is just one experiment. We could repeat the experiment. Maybe the next time you do the experiment you get 14 '1' tickets and 11 '0' tickets, giving a mean of 0.56. Another time you could get 8 '1' tickets and 17 '0' tickets, giving a mean of 0.32.

Task 1 - Your turn! To the right, repeat the experiments over and over again. Every time you do a new experiment, its sample mean will be added to the histogram of the means observed so far. What shape do we see when considering the distribution of the sample means?

Now, let's repeat the task that we did before (repeatedly forming samples) for a new box model that is mostly identical to the one before, except now, each sample consists of only 5 draws from the box.

Task 2 - Your turn! Repeat what we did before, but for when we only take 5 draws from the box. Do we still see a normal distribution?

What is the central limit theorem? The Central Limit Theorem (CLT) states that if you take many random samples of the same size from any population, the distribution of the sample means (or sums) will tend to look like a normal distribution as the sample size gets larger — even if the original population is not normally distributed.

What does this mean for the box model? When we apply the CLT to the box model, it is worth being clear about which distribution we are talking about, because there are three of them, and only one of them is claimed to be normal:

1. The box — this is our population. The tickets in the box are every value that a single draw could produce, in the proportions that they occur. Our box of '1', '0', '0', '0' is a population that is 25% ones and 75% zeros. Nothing about the population changes when we sample from it.
2. One sample. A single experiment: n draws from the box. For n = 25, that is one set of 25 tickets, which gives us one sample mean.
3. The distribution of the sample means. The histogram that you built up by pressing 'Repeat'. Each bar counts how often a sample mean landed in that range, across many repeated experiments.

The CLT is a statement about the third one only. It says that provided each sample is formed from a sufficiently large number of draws (n), the distribution of the sample means (or sums) across repeated samples will be approximately normal. It says nothing about the draws within a single sample — those always just look like the tickets in the box, no matter how large n is.

This is worth pausing on. Our box looks nothing like a normal distribution — it has only two possible values, and it is lopsided (three '0' tickets for every '1'). Yet the histogram of sample means for n = 25 was clearly bell shaped. The normal shape is not inherited from the box; it emerges from the averaging.

How large is 'sufficiently large'? The key words here are sufficiently large. This means that our number of draws (n) must be relatively high. How high? This depends on the tickets in the box.

For example, before we had two box models with the tickets '1', '0', '0', '0', with the only difference being that one box model had samples formed from 25 draws from the box (i.e. n = 25), with the other formed from 5 draws from the box (i.e. n = 5). From the demonstration, we observed that when we repeatedly drew from the box 25 times and calculated the mean, the distribution of the means appeared to be normally distributed. Hence, 25 draws from the box was sufficiently large for the sample means to be normally distributed. However, when we repeated the experiment of drawing from the box 5 times and taking the mean, the distribution was not normally distributed. Hence, five draws from the box was not sufficiently large for the sample means to be normally distributed. In this example, we could have also calculated the sample sums instead. We would have seen the same-looking sample distributions for the n = 25 and n = 5 box models.

You may be wondering if there is some set number of draws (i.e. some threshold value for n) that you can use to say that the CLT always applies. Some people may say that if you have greater than 35 draws, then the CLT will hold, but this is not always the case. You should always look at your underlying data. If the tickets in the box model are symmetric and already nearly normally distributed, then you will not need many draws for the CLT to apply. However, if the box were very skewed (imagine a box with one '1' ticket and ninety '0' tickets), you would likely need many more than 35 draws for the CLT to apply.

In the activity below, you have the opportunity to specify your own box (as well as some boxes that we specified), to investigate how many draws are needed for the sample means (or sums) to be normally distributed.

Task 3 - Your turn!
In this demonstration, you will be changing the contents of the box (that is, the tickets in the box). From this box, we will draw samples of size n = 5, 25, 50 and 100. For each of these sample sizes, you will generate 10,000 samples by pressing the 'SIMULATE' button to the right. You will then look at the different histograms for each sample size to gauge at what point our sample size was large enough for the central limit theorem to apply.

You are free to set your own tickets to place in the box, but we recommend using the practice boxes first, as they include text that describes what we see to help guide you.

Note: Due to how the bin widths are chosen in this exercise, sometimes the distribution of the means and the sums do not match up. For example, for the example 1 box, it appears that the CLT applies for the sample sums for n = 25, but not for the mean when n = 25. You should not be concerned about this difference for this demonstration. In real life (where you have more control), you would experiment with the histogram bin widths until you feel they accurately represent the data.

Current box:

Box contents: