Central Limit Theorem

10 Oct


Many real-world observations can be approximated by, and tested against, the same expected pattern: the normal distribution. In this familiar symmetric bell-shaped pattern, most observations are close to average, and there are fewer observations further from the average. The size of flowers, the physiological response to a drug, the breaking force in a batch of steel cables — these and other observations often fit a normal distribution.

There are, however, many important things we would like to measure and test that do not follow a normal distribution. Household income doesn’t — high values are much further from the average than low values are.

But even when raw data does not fit a normal distribution, there is often a normal distribution lurking within it. This makes it possible to still use the normal distribution to test ideas about non-normal data. This hidden normal distribution is revealed by collecting samples of multiple observations, and calculating the average for each sample. As the number of observations in each sample increases, the distribution of these averages becomes more and more similar to the normal distribution. The existence of this hidden normal distribution is known as the Central Limit Theorem.


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