Statistical regularities had been observed and quantified as early as the mid-18th century by leading mathematicians such as de Moivre, Laplace and Gauss, but fully rigorous formulations and convincing proofs of these properties did not emerge until the 20th century. The laws of large numbers are often reduced to the so-called "central limit theorem" alone. The name was coined by the mathematician George Pólya, who used it in a 1920 article entitled "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung", literally "On the central limit theorem of probability theory". What does this theorem say? It concerns what is known as "convergence in distribution"—that is, convergence of probability distributions. The result concerns the probability distributions of random variables, not the variables themselves!
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The normal distribution
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The normal distribution has the convenient property of being stable under addition, which makes it easy to tabulate: every linear combination of independent normally distributed variables is also normally distributed. The converse of this theorem holds: if a sum of independent random variables is normal, then every term in the sum is normal as well! But there is much more. Every sum of independent, identically distributed variables (that is, variables following the same probability distribution, abbreviated "i.i.d." for "independent and identically distributed") with finite variances, though not necessarily normal ones, is asymptotically normal.
