# Central Limit Theorem

According to the central limit theorem, when an analysis is subject to a variety of indeterminate errors, the distribution of results for that analysis will approximate a normal distribution. As the number of sources of indeterminate error increases, the results more closely approximate a normal distribution. The central limit theorem holds true even if the individual sources of indeterminate error are not normally distributed. The chief limitation to the central limit theorem is that the sources of indeterminate error must be independent and of similar magnitude so that no one source of error dominates the final distribution.

An additional feature of the central limit theorem is that a distribution of means for samples drawn from a population with any distribution will closely approximate a normal distribution if the size of the samples is large enough. For example, the figure below shows two distributions for samples drawn from a uniform distribution  in which every value between 0 and 1 occurs with an equal frequency. Each distribution shows the results for 10,000 samples. For samples of size n = 1 (the histogram on the left), the resulting distribution closely approximates the population’s uniform distribution. The distribution of the means for samples of size n = 10 (the histogram on the right), however, closely approximates a normal distribution.