Lesson 28 of 1524
The central limit theorem
A sample mean varies less than a single observation, and the drop is σ / √n.
Practice this chapterThe sample mean is a random variable. Its sampling distribution is centered at the population mean μ. Its standard deviation is the standard error, σ / √n.
If the population is normal, the sample mean is normal for every sample size. If it is not, the central limit theorem says the sample mean is still approximately normal once n is large. This text often uses 30 as a guide.
Standard error of the mean
σₓ̄ = σ / √n
Larger samples make the mean less variable.
Worked example
σ = 20 and n = 16. What is the standard error of the mean?
- 1The standard error of the sample mean is σ / √n. The divisor is the square root of the sample size, not the sample size itself.
- 2√16 = 4, because 4 × 4 = 16.
- 3Divide the population standard deviation by that root: 20 / 4 = 5.
- 4A single observation typically sits about 20 units from μ. A mean of 16 observations typically sits about 5 units from μ, because the observations partly cancel.
Result: 5
Why. Averaging 16 observations cuts the typical distance from the mean by √16, which is 4. So 20 becomes 20 / 4 = 5. Dividing by 16 instead of by 4 would give 1.25, and that is too small.
Dividing by n, instead of by √n, makes the standard error much too small.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.