Which statement best describes the Central Limit Theorem in one line?

Which statement best describes the Central Limit Theorem in one line?

• A. The sum (or average) of i.i.d. random variables with finite mean and variance converges to a normal distribution as sample size grows.
• B. The distribution of the maximum of i.i.d. variables converges to a normal distribution.
• C. The sample mean of independent variables becomes exactly normal for any sample size.
• D. The sum of independent variables is always normal.

Answer: (A)

The Central Limit Theorem describes how the sum (or average) of many independent, identically distributed random variables with finite mean and variance becomes approximately normally distributed as the number of terms grows. In more detail, if the Xi are i.i.d. with mean μ and variance σ^2, then the standardized sum (S_n − nμ)/(√n σ) converges in distribution to a standard normal as n increases, which means the sample mean X̄_n = S_n/n tends to N(μ, σ^2/n). So for large sample sizes, the sum or average is well approximated by a normal distribution, regardless of the original distribution. The idea that the maximum converges to a normal distribution is not correct, as extremes follow different limiting behaviors. Also, the sample mean being exactly normal for any sample size isn’t true; it’s only exactly normal in special cases (for example, if the original distribution is normal), while in general it’s an approximation that improves with larger n. Similarly, the sum of independent variables isn’t always normal—the normality emerges in the large-sample limit under the stated conditions.

The Central Limit Theorem describes how the sum (or average) of many independent, identically distributed random variables with finite mean and variance becomes approximately normally distributed as the number of terms grows. In more detail, if the Xi are i.i.d. with mean μ and variance σ^2, then the standardized sum (S_n − nμ)/(√n σ) converges in distribution to a standard normal as n increases, which means the sample mean X̄_n = S_n/n tends to N(μ, σ^2/n). So for large sample sizes, the sum or average is well approximated by a normal distribution, regardless of the original distribution. The idea that the maximum converges to a normal distribution is not correct, as extremes follow different limiting behaviors. Also, the sample mean being exactly normal for any sample size isn’t true; it’s only exactly normal in special cases (for example, if the original distribution is normal), while in general it’s an approximation that improves with larger n. Similarly, the sum of independent variables isn’t always normal—the normality emerges in the large-sample limit under the stated conditions.