Chapter 3 The Central Limit Theorem

We now turn our attention to one of the most fundamental results in statistics: The remarkable Central Limit Theorem.

The Central Limit Theorem (CLT)

Let \(X_1, \ldots, X_n\) be a random sample from a distribution with finite mean \(\mu\) and finite variance \(\sigma^2\). For \(\overline{X}\) denoting the sample mean, if \(n\) is sufficiently large then \[\overline{X}\stackrel{\tiny \text{approx.}}\sim N\left(\mu,\frac{\sigma^2}{n}\right)\] where \(\stackrel{\tiny \text{approx.}}\sim\) denotes 'approximately distributed as'.

Normally, a sample size of approximately \(n = 30\) is considered to be 'sufficiently large'.

To help us understand this theorem, let's consider some simulated examples.

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND