> In probability theory, the central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of independent random va

> In probability theory, the central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed (i.i.d.). No, that's just wrong. The context is some positive integer n and values of some n real valued random variables independent and identically distributed (i.i.d.). If for some n do what the quote says, then are not guaranteed to get a Gaussian. Moreover, if do what the quote says and take the limit as n goes to positive infinity, then, say, in the case the random variables have expectation that is finite and finite variance, will get just the common expectation of those random variables -- this is the law of large numbers, not the central limit theorem. That the distribution of the mean converges to the point with the expectation is the 'weak' law of large numbers. That the mean converges to the expectation with probability 1 is the strong law of large numbers. For the central limit theorem, add up the n i.i.d. values and divide by the square root of n. Then under meager assumptions (e.g., i.i.d. and, for likely the strongest conditions known, the Lindeberg-Feller conditions), as n goes to infinity, the distribution of the result will converge to a Gaussian ('normal') distribution. Authors with the details include J. Neveu, L. Breiman, M. Loeve, and K. Chung. It's essential to divide by the square root of n, not just n.