Which formula represents the calculation for the standard deviation of x̅ under the Central Limit Theorem?

The calculation for the standard deviation of the sample mean (x̅) under the Central Limit Theorem is given by the formula that divides the population standard deviation by the square root of the sample size. This is referred to as the standard error of the mean.

This concept is fundamental because it shows how the variability of the sample mean decreases as the sample size increases. Specifically, as the value of N (the sample size) grows larger, the denominator (which includes the square root of N) increases, leading to a smaller standard error. This means that larger samples tend to provide a more accurate estimate of the population mean due to the reduced variability in the sample means.

Using the standard deviation of the population in this context ensures that you are accurately reflecting the distribution of sample means, which is essential for making inferences about the population based on sample data. This relationship underscores the Central Limit Theorem, which states that, given a sufficiently large sample size, the distribution of the sample means will approximate a normal distribution, regardless of the distribution of the population.