Central Limit Theorem

When you repeat an experiment over and over and add the results, the probability distributions change in an interesting way.

One Six-Sided Die

This distribution doesn’t look anything like a bell curve, but it does have a mean ($\mu$) and a standard deviation ($\sigma$):

• $\mu = 3.5$
• $\sigma = 1.7078$

Four Dice

Now we have something that looks a lot like a bell curve. Here are the mean and standard deviations for $\bar{x}$.

• $\mu_{\bar{x}} = 3.5$
• $\displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{4}} = \frac{1.7078}{2} = 0.85391$

Notice that the middle hasn’t changed, but the results are only half as spread out.

Ten Dice

• $\mu_{\bar{x}} = 3.5$
• $\displaystyle \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{10}} = \frac{1.7078}{3.1623} = 0.54006$

Thirty Dice

Calculate $\mu$ and $\sigma$ for the average result of 30 dice.