Which concept states that the distribution of sample averages tends to be normal regardless of the shape of the population distribution?

Which concept states that the distribution of sample averages tends to be normal regardless of the shape of the population distribution?

• A. Central Limit Theorem
• B. Law of Large Numbers
• C. Normal distribution
• D. Standard error

Answer: (A)

Central Limit Theorem: as sample size grows, the distribution of the sample mean becomes approximately normal, even if the underlying population distribution is not normal, as long as the population variance is finite. This happens because averaging many independent random contributions smooths out irregularities, yielding a bell-shaped sampling distribution centered at the true population mean. The spread of that distribution is the standard error, which equals the population standard deviation divided by the square root of the sample size; it shrinks as the sample size increases, making the normal pattern more pronounced. In practice, this justifies using normal-approximation methods for means with reasonably large samples, regardless of the population’s shape. The law of large numbers describes the sample mean converging to the population mean with increasing n, not the normal shape of the sampling distribution. A normal distribution is the form the sampling distribution takes under the CLT, not a separate rule about all distributions. The standard error is the measure of that distribution’s spread, not an explanation for why it becomes normal.

Central Limit Theorem: as sample size grows, the distribution of the sample mean becomes approximately normal, even if the underlying population distribution is not normal, as long as the population variance is finite. This happens because averaging many independent random contributions smooths out irregularities, yielding a bell-shaped sampling distribution centered at the true population mean. The spread of that distribution is the standard error, which equals the population standard deviation divided by the square root of the sample size; it shrinks as the sample size increases, making the normal pattern more pronounced. In practice, this justifies using normal-approximation methods for means with reasonably large samples, regardless of the population’s shape. The law of large numbers describes the sample mean converging to the population mean with increasing n, not the normal shape of the sampling distribution. A normal distribution is the form the sampling distribution takes under the CLT, not a separate rule about all distributions. The standard error is the measure of that distribution’s spread, not an explanation for why it becomes normal.