Central limit theorem

The central limit theorem is a theorem in probability theory that states that if you have several independent samples from a probability distribution with finite variance, then as the sample size approaches infinity, the sum and average of the samples approach a normal distribution. Formally, it says that given a sequence of independent, identically distributed real-valued random variables $$X_1, X_2, X_3, \dots$$ with mean $$\mu$$ and variance $$\sigma^2 < \infty$$, if $$\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n}X_i$$ is the mean of the first n variables in the sequence, then $$\sqrt{n}(\bar{X}_n - \mu)$$ converges to $$N(0, \sigma^2)$$ in distribution as $$n \rightarrow \infty$$. As a consequence, if n is large enough, $$N(\mu, \frac{\sigma^2}{n})$$ will be a good approximation of the distribution of the sample mean even if the distribution of X is not close to normal.

Applications
The central limit theorem is often used to justify applying certain statistical tests to analyze experimental data. One of the earliest forms of the central limit theorem, the De Moivre-Laplace theorem, applied specifically to Bernoulli (binary) random variables. Today, experiments often use z-tests to compare the frequencies of binary outcomes. Margins of error in surveys are often based on the normal approximation justified by the central limit theorem.

Another common statistical test for analyzing experimental data is called the which is used to assess whether two independent sets of quantitative data have the same mean, or whether the mean of one set of data is equal to a predetermined value. One of the assumptions required for the t-test to be valid is that the data is normally distributed, but the central limit theorem allows this requirement to be relaxed if the datasets are large enough. A t-test statistic is a (continuous) function of the sample mean and sample variance of the dataset(s) involved in the test. The central limit theorem indicates that if the datasets are large enough and the data meets the assumptions of the central limit theorem, then the sample mean will be close to normal, as it would be with normally distributed data. In addition, since the observations in a dataset that meets the assumptions of the central limit theorem have finite variance, due to the law of large numbers. Therefore, the distribution of the t-test statistic using non-normally distributed data that meets the conditions of the central limit theorem will be similar to that of normally distributed data for large sample sizes, and so the t-test can often be applied to quantitative data without first checking normality if the sample size is large enough.

Use to "prove" the existence of God
Abraham de Moivre proved the central limit theorem for the specific case of Bern(0.5), a binary random variable in which each outcome is equally likely (e.g. tossing a fair coin). He believed that this represented a proof of the existence of God, on the basis that it shows that even random events ultimately converge to a predictable outcome over time, and so chance is not a threat to God's intentions.

There are several problems with this reasoning. First, the central limit theorem only applies in the limit as the number of chance outcomes approaches infinity. Phenomena in the real world will depend on finite chance events, and sometimes may not even depend on a large enough number of outcomes to apply the central limit theorem to. The normal approximation implied by the central limit theorem may be a good model for the annual rainfall in your city, but the weather tomorrow is still governed by chaos.

Second, the central limit theorem does not apply to every distribution. For example, the sample mean of any number of independent realizations from the (the quotient of two independent standard normal random variables) will also have a Cauchy distribution. If the central limit theorem were a demonstration of divine omnipotence, then that divine omnipotence would have exceptions. De Moivre himself had only proven the theorem for one specific case, meaning he had even less justification to regard it as proof of divinity.

Finally, even if the central limit theorem is regarded as showing that omnipotence is consistent with chance, it doesn't mean that an omnipotent deity must exist.