Which statement describes the relationship between sample size and the standard error of the mean?

Which statement describes the relationship between sample size and the standard error of the mean?

• A. The standard error increases with sample size.
• B. The standard error is unaffected by sample size.
• C. The standard error decreases and then increases as sample size grows.
• D. The standard error decreases as sample size increases.

Answer: (D)

The standard error of the mean measures how precisely the sample mean estimates the population mean. It shrinks as you collect more data because averaging more observations reduces the random fluctuations that cause the sample mean to vary from sample to sample. Mathematically, the standard error is the population standard deviation divided by the square root of the sample size (SE = σ/√n, or s/√n when using the sample standard deviation). This means that increasing the sample size makes the standard error smaller, with doubling the size reducing the standard error by a factor of about 1/√2 and quadrupling it reducing it by half. Therefore, the standard error decreases as the sample size increases.

The standard error of the mean measures how precisely the sample mean estimates the population mean. It shrinks as you collect more data because averaging more observations reduces the random fluctuations that cause the sample mean to vary from sample to sample. Mathematically, the standard error is the population standard deviation divided by the square root of the sample size (SE = σ/√n, or s/√n when using the sample standard deviation). This means that increasing the sample size makes the standard error smaller, with doubling the size reducing the standard error by a factor of about 1/√2 and quadrupling it reducing it by half. Therefore, the standard error decreases as the sample size increases.