The Central Limit Theorem describes the characteristics of the "

**population of the means"**which has been created from the means of an infinite number of random population samples of size (N), all of them drawn from a given "**parent population**". The Central Limit Theorem predicts that__regardless of the distribution of the parent population__:**[1]**The

**mean**of the population of means is always equal to the mean of the parent population from which the population samples were drawn.

**[2]**The

**standard deviation**of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N).

**[3]**[

*] The distribution of means will increasingly approximate a*

**And the most amazing part!!****normal distribution**as the size N of samples increases.

A consequence of Central Limit Theorem is that if we average measurements of a particular quantity, the distribution of our average tends toward a normal one. In addition, if a measured variable is actually a combination of several other uncorrelated variables, all of them "contaminated" with a random error of any distribution, our measurements tend to be contaminated with a random error that is normally distributed as the number of these variables increases.

Thus, the Central Limit Theorem explains the ubiquity of the famous bell-shaped "Normal distribution" (or "Gaussian distribution") in the measurements domain.

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