What is the standard deviation of the sample mean x̅ approximately equal to?

The standard deviation of the sample mean, often referred to as the standard error of the mean, is calculated using the formula for the standard deviation divided by the square root of the sample size. This relationship arises from the central limit theorem, which asserts that the distribution of the sample mean will approximate a normal distribution as the sample size increases, regardless of the population's distribution.

When you take a sample of size N from a population with a certain standard deviation, the variability of the sample means is less than that of the individual observations. Specifically, by dividing the population standard deviation by the square root of the sample size (N), you adjust for the fact that larger samples tend to yield more accurate and less variable estimates of the population mean. This reflects the principle that as you increase the sample size, the precision of your estimate improves, leading to a smaller standard deviation of the sample mean.

In summary, the standard deviation of the sample mean is approximately equal to the standard deviation of the population divided by the square root of the sample size, confirming that the answer choice relating to this is correct.