Central Limit Theorem

The third property of the sampling distribution of sample means pertains to the shape of the distribution...

The Central Limit Theorem

As the sample size \(n\) increases without limit, the shape of the distribution of sample means taken with replacement from a population with mean \(\mu\) and standard deviation \(\sigma\) will approach a normal distribution.

Some important facts about the CLT:

When the original variable is normally distributed, the distribution of the sample means will be normally distributed, for any sample size \(n\).
When the distribution of the original variable is not normal, a sample size of 30 or more is needed to use a normal distribution to approximate the distribution of the sample means. The larger the sample, the better the approximation will be.

Converting \(\overline{x}\) to \(z\)-scores:

\[z=\frac{\overline{x}-\mu}{\Big(\frac{\sigma}{\sqrt{n}}\Big)}=\frac{\sqrt{n}\big(\overline{x}-\mu\big)}{\sigma}\]