What is the standard error of the mean and its formula?

What is the standard error of the mean and its formula?

• A. The standard deviation of the sampling distribution of the sample mean: SE = σ/√n.
• B. The population standard deviation.
• C. The difference between sample and population means.
• D. The standard deviation of the population.

Answer: (C)

Think about taking many samples of size n from a population and computing the mean for each sample. Those sample means form a distribution whose spread is called the standard error of the mean. If you know the population standard deviation, this spread is SE = σ/√n. When σ isn’t known, you estimate it with the sample standard deviation and use SE ≈ s/√n. This shows why larger samples yield more precise estimates of the population mean—the variability of the sample mean across samples decreases as n grows. The other ideas describe different concepts: the population standard deviation is the spread of individual values in the population, not the variability of sample means; and the difference between a sample mean and the population mean is an estimation error for a single sample, not the spread of many sample means.

Think about taking many samples of size n from a population and computing the mean for each sample. Those sample means form a distribution whose spread is called the standard error of the mean. If you know the population standard deviation, this spread is SE = σ/√n. When σ isn’t known, you estimate it with the sample standard deviation and use SE ≈ s/√n. This shows why larger samples yield more precise estimates of the population mean—the variability of the sample mean across samples decreases as n grows. The other ideas describe different concepts: the population standard deviation is the spread of individual values in the population, not the variability of sample means; and the difference between a sample mean and the population mean is an estimation error for a single sample, not the spread of many sample means.