Which statement about the standard error of the mean is true?

Which statement about the standard error of the mean is true?

• A. It increases with sample size.
• B. It equals the population standard deviation.
• C. It equals the population standard deviation divided by the square root of the sample size.
• D. It is equal to the square root of the sample variance.

Answer: (C)

The standard error of the mean captures how much the sample average is expected to vary from one random sample to another. When you take independent observations with a constant variance, the sampling distribution of the sample mean has a standard deviation equal to the population standard deviation divided by the square root of the sample size. In formula terms, SE(mean) = sigma / sqrt(n). This means larger samples give more precise estimates of the true mean, since the variability of the sample mean decreases with n. You can estimate this from data by using s / sqrt(n), where s is the sample standard deviation. Why the other ideas don’t fit: the standard error is not the population standard deviation itself, and it’s not the square root of the sample variance (that would be the sample standard deviation, not the standard error of the mean). Also, the standard error decreases as sample size grows, it does not increase.

The standard error of the mean captures how much the sample average is expected to vary from one random sample to another. When you take independent observations with a constant variance, the sampling distribution of the sample mean has a standard deviation equal to the population standard deviation divided by the square root of the sample size. In formula terms, SE(mean) = sigma / sqrt(n). This means larger samples give more precise estimates of the true mean, since the variability of the sample mean decreases with n. You can estimate this from data by using s / sqrt(n), where s is the sample standard deviation.

Why the other ideas don’t fit: the standard error is not the population standard deviation itself, and it’s not the square root of the sample variance (that would be the sample standard deviation, not the standard error of the mean). Also, the standard error decreases as sample size grows, it does not increase.