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The Central Limit Theorem: Convergence of Sample Means to a Normal Distribution

The Central Limit Theorem (CLT) establishes that for any independent and identically distributed population with finite variance, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases. This theorem demonstrates that repeated averaging of random variables from non-normal distributions results in convergence to Gaussian parameters, fundamentally linking arbitrary probability spaces to the theory of estimation via standard error analysis within statistics.