The law of large numbers states that the sample mean of a sequence of independent, identically distributed random variables converges to the true (theoretical) mean as sample size grows—almost surely in its strong form, and in probability in its weak form (provable via Chebyshev's inequality)—establishing that averaging over increasing amounts of data reliably recovers an underlying population parameter. The central limit theorem sharpens this by describing the rate and shape of that convergence: the sample mean, standardized by subtracting its mean and dividing by its standard deviation scaled by the square root of n, converges in distribution to the standard normal distribution regardless of the underlying distribution's shape, provided it has finite variance. Both results belong to probability theory's asymptotic theory of sums of independent random variables, underpinning statistical estimation and the widespread use of normal approximations, including for the binomial distribution.
The law of large numbers states that the sample mean of a sequence of independent, identically distributed random variables converges to the true (theoretical) mean as sample size grows—almost surely…