# The Central Limit Theorem and its Implications for Statistical Inference

Jeremy Albright

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central limit theorem logic

## Logic

The central limit theorem is perhaps the most fundamental result in all of statistics. It allows us to understand the behavior of estimates across repeated sampling and thereby conclude if a result from a given sample can be declared to be “statistically significant,” that is, different from some null hypothesized value. This brief tutorial explains what the central theorem tells us and why the result is important for statistical inference.

The central limit theorem tells us exactly what the shape of the distribution of means will be when we draw repeated samples from a given population. Specifically, as the sample sizes get larger, the distribution of means calculated from repeated sampling will approach normality. What makes the central limit theorem so remarkable is that this result holds no matter what shape the original population distribution may have been.

As an example, say that we find a school that has 1200 students, with exactly 200 students each in grades 7 through 12. The population distribution is, as the following figure shows, definitely not normal.