State the Central Limit Theorem in one line.

State the Central Limit Theorem in one line.

• A. The law of large numbers states the sample average converges to the population mean as n increases.
• B. The sum of independent random variables is always normally distributed.
• C. The sum (or average) of i.i.d. random variables with finite mean and variance converges to a normal distribution as sample size grows.
• D. The central limit theorem applies only to normal populations.

Answer: (C)

The central idea is that the sum (or the average) of a large number of independent, identically distributed random variables with finite mean and finite variance behaves like a normal distribution as the sample size grows. In more precise terms, if S_n is the sum and mu and sigma^2 are the mean and variance of the individual draws, then the standardized sum (S_n - n mu) / (sigma sqrt(n)) converges in distribution to a standard normal as n becomes large. This means that, regardless of the original shape of the distribution, the distribution of the sum (or average) becomes approximately normal for large n, after proper standardization. This is exactly what the correct statement captures: the sum (or average) of i.i.d. random variables with finite mean and variance converges to a normal distribution as sample size grows. The other options mix up what the theorem implies—the law of large numbers describes convergence of the mean itself, not distributional convergence to normal; the sum is not always normal unless the original variables are normal; and the theorem is not restricted to normal populations.

The central idea is that the sum (or the average) of a large number of independent, identically distributed random variables with finite mean and finite variance behaves like a normal distribution as the sample size grows. In more precise terms, if S_n is the sum and mu and sigma^2 are the mean and variance of the individual draws, then the standardized sum (S_n - n mu) / (sigma sqrt(n)) converges in distribution to a standard normal as n becomes large. This means that, regardless of the original shape of the distribution, the distribution of the sum (or average) becomes approximately normal for large n, after proper standardization. This is exactly what the correct statement captures: the sum (or average) of i.i.d. random variables with finite mean and variance converges to a normal distribution as sample size grows. The other options mix up what the theorem implies—the law of large numbers describes convergence of the mean itself, not distributional convergence to normal; the sum is not always normal unless the original variables are normal; and the theorem is not restricted to normal populations.