Worksheets · Statistics

Sampling distributions worksheet

A sample proportion p̂ is centered on the population p with standard deviation √(p(1 − p)/n). A sample mean x̄ is centered on μ with standard deviation σ/√n. Bigger samples vary less.

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Name

Sampling Distributions

Date Period

Find the mean and standard deviation of each sampling distribution.

  1. A population has mean 115 and standard deviation 12. For random samples of 36, find the mean and standard deviation of xˉ
  2. A population has mean 70 and standard deviation 14. For random samples of 49, find the mean and standard deviation of xˉ
  3. A population has mean 95 and standard deviation 9. For random samples of 81, find the mean and standard deviation of xˉ
  4. A population has mean 110 and standard deviation 8. For random samples of 16, find the mean and standard deviation of xˉ
  5. 50% of a large population has a trait. For random samples of 25, find the mean and standard deviation of p^
  6. 50% of a large population has a trait. For random samples of 100, find the mean and standard deviation of p^
  7. 40% of a large population has a trait. For random samples of 96, find the mean and standard deviation of p^
  8. 90% of a large population has a trait. For random samples of 100, find the mean and standard deviation of p^
  9. A population has mean 70 and standard deviation 6. For random samples of 4, find the mean and standard deviation of xˉ
  10. 20% of a large population has a trait. For random samples of 16, find the mean and standard deviation of p^
  11. A population has mean 55 and standard deviation 6. For random samples of 4, find the mean and standard deviation of xˉ
  12. A population has mean 160 and standard deviation 8. For random samples of 16, find the mean and standard deviation of xˉ
  13. A population has mean 160 and standard deviation 10. For random samples of 25, find the mean and standard deviation of xˉ
  14. 20% of a large population has a trait. For random samples of 64, find the mean and standard deviation of p^
  15. A population has mean 65 and standard deviation 10. For random samples of 25, find the mean and standard deviation of xˉ
  16. A population has mean 200 and standard deviation 10. For random samples of 25, find the mean and standard deviation of xˉ
  17. A population has mean 75 and standard deviation 8. For random samples of 16, find the mean and standard deviation of xˉ
  18. A population has mean 190 and standard deviation 14. For random samples of 49, find the mean and standard deviation of xˉ
  19. A population has mean 170 and standard deviation 14. For random samples of 49, find the mean and standard deviation of xˉ
  20. A population has mean 25 and standard deviation 12. For random samples of 36, find the mean and standard deviation of xˉ

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sampling DistributionsAnswer keyVersion 1

Date Period

Find the mean and standard deviation of each sampling distribution.

  1. A population has mean 115 and standard deviation 12. For random samples of 36, find the mean and standard deviation of xˉmean 115, standard deviation 2
  2. A population has mean 70 and standard deviation 14. For random samples of 49, find the mean and standard deviation of xˉmean 70, standard deviation 2
  3. A population has mean 95 and standard deviation 9. For random samples of 81, find the mean and standard deviation of xˉmean 95, standard deviation 1
  4. A population has mean 110 and standard deviation 8. For random samples of 16, find the mean and standard deviation of xˉmean 110, standard deviation 2
  5. 50% of a large population has a trait. For random samples of 25, find the mean and standard deviation of p^mean 0.5, standard deviation 0.1
  6. 50% of a large population has a trait. For random samples of 100, find the mean and standard deviation of p^mean 0.5, standard deviation 0.05
  7. 40% of a large population has a trait. For random samples of 96, find the mean and standard deviation of p^mean 0.4, standard deviation 0.05
  8. 90% of a large population has a trait. For random samples of 100, find the mean and standard deviation of p^mean 0.9, standard deviation 0.03
  9. A population has mean 70 and standard deviation 6. For random samples of 4, find the mean and standard deviation of xˉmean 70, standard deviation 3
  10. 20% of a large population has a trait. For random samples of 16, find the mean and standard deviation of p^mean 0.2, standard deviation 0.1
  11. A population has mean 55 and standard deviation 6. For random samples of 4, find the mean and standard deviation of xˉmean 55, standard deviation 3
  12. A population has mean 160 and standard deviation 8. For random samples of 16, find the mean and standard deviation of xˉmean 160, standard deviation 2
  13. A population has mean 160 and standard deviation 10. For random samples of 25, find the mean and standard deviation of xˉmean 160, standard deviation 2
  14. 20% of a large population has a trait. For random samples of 64, find the mean and standard deviation of p^mean 0.2, standard deviation 0.05
  15. A population has mean 65 and standard deviation 10. For random samples of 25, find the mean and standard deviation of xˉmean 65, standard deviation 2
  16. A population has mean 200 and standard deviation 10. For random samples of 25, find the mean and standard deviation of xˉmean 200, standard deviation 2
  17. A population has mean 75 and standard deviation 8. For random samples of 16, find the mean and standard deviation of xˉmean 75, standard deviation 2
  18. A population has mean 190 and standard deviation 14. For random samples of 49, find the mean and standard deviation of xˉmean 190, standard deviation 2
  19. A population has mean 170 and standard deviation 14. For random samples of 49, find the mean and standard deviation of xˉmean 170, standard deviation 2
  20. A population has mean 25 and standard deviation 12. For random samples of 36, find the mean and standard deviation of xˉmean 25, standard deviation 2

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

A population has mean 50 and standard deviation 12. For samples of 36, find the mean and standard deviation of xˉ

  1. The mean of x̄ is the population mean, 50.
  2. Its standard deviation is 12/√36 = 2.

The answer is mean 50, standard deviation 2.

Where students go wrong

Dividing by n instead of √n. Four times the sample size halves the spread; it does not quarter it.

Also called sampling distribution of p-hat, sampling distribution of x-bar or standard error.

Questions about these worksheets

Yes. Every sampling distributions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Statistics is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the sampling distributions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Dividing by n instead of √n. Four times the sample size halves the spread; it does not quarter it.