A probabilistic result stating that suitably normalized sums (or averages) of many independent, identically distributed random variables with finite variance converge in distribution to a Gaussian (normal) law as the number of terms grows without bound.

Principle

Aggregation and normalization transform microscopic randomness into a universal, stable attractor (the Gaussian) under mild regularity conditions.

Demonstration

Compute sample means of size n from a uniform[0,1] distribution for increasing n; the histogram of normalized sums approaches the bell curve shape of the normal density.

Misapplication

Applying the theorem to sums of heavy-tailed variables with infinite variance; such sums may converge to a stable non-Gaussian law instead.

Consequence

Justifies normal approximations in inference and error propagation for large samples, enabling z-based confidence intervals and hypothesis tests when conditions hold.

Reversal

Without aggregation or when variance is infinite, the limit behavior retains the original distributional features or follows a different stable law rather than Gaussianity.

Boundary

Requires independence or weak dependence and finite second moment; does not apply for strongly dependent sequences, very small samples, or distributions lacking finite variance.

Semantic Tension

Competes with finite-sample exact distributions and with stable-law limits for heavy tails; practical tension arises between asymptotic Gaussian approximations and small-sample accuracy.

Synthesis

The central limit theorem is the statement that under weak regularity the normalized sum of many independent random contributions converges to a Gaussian law, providing a universal approximation for aggregate behavior.