The Central Limit Theorem states that

The Central Limit Theorem states that

• A. The sum of a large number of independent, identically distributed random variables is exactly normal.
• B. The sum (or average) of a large number of independent, identically distributed variables with finite mean and variance is approximately normally distributed.
• C. Only normal variables follow the theorem.
• D. The variance of the sum becomes zero as the number of terms grows.

Answer: (D)

The central limit behavior comes from how many independent little effects add up. If you have a large number of independent, identically distributed random variables with a finite mean and finite variance, the distribution of their sum (and, after scaling as an average, the distribution of the mean) tends to a normal shape as the sample size grows. It’s an approximation that gets better with more terms, not an exact rule in general. The variance of the sum itself grows with the number of terms (it’s proportional to n), while the variance of the average shrinks like 1/n. So the best description is that the sum or average is approximately normally distributed for large n, rather than exactly normal or only for normal variables. This is why, in practice, we often invoke normality for large samples when making inferences about means.

The central limit behavior comes from how many independent little effects add up. If you have a large number of independent, identically distributed random variables with a finite mean and finite variance, the distribution of their sum (and, after scaling as an average, the distribution of the mean) tends to a normal shape as the sample size grows. It’s an approximation that gets better with more terms, not an exact rule in general. The variance of the sum itself grows with the number of terms (it’s proportional to n), while the variance of the average shrinks like 1/n. So the best description is that the sum or average is approximately normally distributed for large n, rather than exactly normal or only for normal variables. This is why, in practice, we often invoke normality for large samples when making inferences about means.