# Random sampling
Random sampling selects units from a defined population through a chance mechanism in which every unit has a known probability of selection, and it underpins survey estimation and design-based statistical inference. Only a random draw allows sample results to be generalized to the whole population with calculable margins of error.[1](https://www.webpages.uidaho.edu/population_ecology/Simple_random_sampling.htm) • [2](https://sda.berkeley.edu/dlab/sampling/Piazza_draft.pdf) Simple random sampling (SRS) is the standard basic method and serves as the baseline against which other designs are evaluated.[2](https://sda.berkeley.edu/dlab/sampling/Piazza_draft.pdf) One caution applies from the outset: giving every unit the same chance of selection is not by itself sufficient; a simple random sample requires that all possible samples of a given size are equally likely.[3](https://www.amsi.org.au/ESA_Senior_Years/PDF/RandomSampling4b.pdf)
| Key fact | Detail |
|---|---|
| What is selected | Units from a sampling frame, such that every unit has a known chance of selection[1](https://www.webpages.uidaho.edu/population_ecology/Simple_random_sampling.htm) • [3](https://www.amsi.org.au/ESA_Senior_Years/PDF/RandomSampling4b.pdf) |
| Inclusion probability (SRSWOR) | \( \pi_{i} = n/N \) for every unit; joint \( \pi_{ij} = n(n-1)/[N(N-1)] \)[4](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html) |
| Finite population correction | \( \text{fpc} = 1 - n/N \); approximately 1 for small sampling fractions, 0 for a census[4](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html) |
| Estimator | Horvitz–Thompson estimator with base weights \( w_{i} = \pi_{i}^{-1} \), unbiased when all inclusion probabilities are known and positive[5](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations) |
| Efficiency | SRS without replacement gives lower estimator variances than SRS with replacement[5](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations) |
| Historical watershed | Neyman's 1934 paper on the two aspects of the representative method[6](https://doi.org/10.2307/2342192) |
| Random numbers in software | Computer random numbers are pseudo-random, generated by a formula[7](https://www.stat.auckland.ac.nz/~wild/ChanceEnc/Ch01.random.pdf); R's sample() results changed with version 3.6.0[8](https://stat.ethz.ch/R-manual/R-devel/library/base/html/sample.html) |
## How it works
A probability sampling procedure must meet four criteria: the distinct samples that can result are definable, the selection probability of each is known, selection is carried out randomly with those probabilities, and each sample yields a unique estimate.[1](https://www.webpages.uidaho.edu/population_ecology/Simple_random_sampling.htm) Under simple random sampling without replacement (SRSWOR) there are \( \binom{N}{n} \) distinct possible samples, each selected with probability \( 1/\binom{N}{n} \), so each unit's inclusion probability is \( \pi_{i} = n/N \).[4](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html) • [9](https://arxiv.org/pdf/1612.04965) With replacement (SRSWR), each unit has chance \( 1/N \) at every draw, giving \( N^{n} \) possible ordered samples each with probability \( 1/N^{n} \), and an inclusion probability of \( 1-\left(1-\frac{1}{N}\right)^{n} \), which is smaller than \( n/N \).[10](https://dickbrus.github.io/SpatialSamplingwithR/IntroProbabilitySampling.html) • [11](https://home.iitk.ac.in/~shalab/sampling/chapter2-sampling-simple-random-sampling.pdf)
The sample mean is unbiased for the population mean, with variance \( \operatorname{Var}(\bar{y}) = (1-n/N) \cdot \frac{S^{2}}{n} \), where \( S^{2} = \sum_{i}(y_{i}-\bar{Y})^{2}/(N-1) \) is the finite-population variance and \( (N-n)/N \) is the finite population correction fraction.[12](https://online.stat.psu.edu/stat506/Lesson01) Under SRSWR the variance is \( (N-1)S^{2}/(Nn) \).[13](https://dept.stat.lsa.umich.edu/~moulib/sampling.pdf) For arbitrary designs, the total is estimated by the Narain–[Horvitz–Thompson estimator](https://www.edgepedia.org/horvitz-thompson-estimator) \( \hat{Y} = \sum_{i}^{n} w_{i} y_{i} \) with base weights \( w_{i} = \pi_{i}^{-1} \), unbiased provided all first-order inclusion probabilities are positive.[5](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations) • [14](https://doi.org/10.1080/01621459.1952.10483446) • [9](https://arxiv.org/pdf/1612.04965) SRSWOR is more efficient than SRSWR because no element can be selected more than once; a reselected unit carries no additional information, though sampling with replacement is easier to implement.[5](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations) • [10](https://dickbrus.github.io/SpatialSamplingwithR/IntroProbabilitySampling.html)
## How it is done
The population must first be defined and a sampling frame constructed, the practical population actually sampled from, which should match the real population of interest as closely as possible.[3](https://www.amsi.org.au/ESA_Senior_Years/PDF/RandomSampling4b.pdf) The N frame units are numbered 1 to N, and random numbers are read from a table or generator: repeats are accepted for SRSWR and ignored for SRSWOR.[11](https://home.iitk.ac.in/~shalab/sampling/chapter2-sampling-simple-random-sampling.pdf) A calculator method converts a uniform random number u to a unit index through \( i = 1 + u \cdot N \), taking the integer part.[7](https://www.stat.auckland.ac.nz/~wild/ChanceEnc/Ch01.random.pdf) A spreadsheet method assigns each element a random number, sorts, and takes the first n elements.[1](https://www.webpages.uidaho.edu/population_ecology/Simple_random_sampling.htm) • [3](https://www.amsi.org.au/ESA_Senior_Years/PDF/RandomSampling4b.pdf) Statistical software such as SPSS, SAS, or Stata can select samples directly from [computer file](https://www.edgepedia.org/computer-file) frames,[2](https://sda.berkeley.edu/dlab/sampling/Piazza_draft.pdf) as can R's sample(1:N, size = n, replace = FALSE), Minitab's Sample from Columns, and Python's random.sample.[8](https://stat.ethz.ch/R-manual/R-devel/library/base/html/sample.html) • [15](https://www.casrai.org/guides/simple-random-sampling) • [16](https://online.stat.psu.edu/stat200/book/export/html/23) Where the draw must be exactly reproducible for a pre-registration or audit trail, a seed is set first with set.seed().[15](https://www.casrai.org/guides/simple-random-sampling) [Reproducibility](https://www.edgepedia.org/reproducibility) now depends on documented seeds and generator settings: R's sample() results changed with version 3.6.0 through RNGkind(sample.kind = ..), a change that breaks reproduction of older scripts.[8](https://stat.ethz.ch/R-manual/R-devel/library/base/html/sample.html)
## Origin
The representative method entered statistics through surveys for a Norwegian retirement and sickness insurance scheme.[17](https://pure.uva.nl/ws/files/24636645/74286_312955.pdf) The monograph described sampling persons from selected parishes and towns, in connection with the 1891 Norwegian census, so the sample would form a correct miniature of the whole population.[18](https://www.ssb.no/a/histstat/sos/sos_027.pdf) • [19](https://projecteuclid.org/journalArticle/Download?urlId=10.1214%2Fss%2F1177013352&isResultClick=False) A practical demonstration was provided in a Reading survey published in 1912, showing that estimates from large random samples have an approximately normal distribution.[20](https://www150.statcan.gc.ca/n1/pub/12-001-x/2013002/article/11883-eng.htm) • [17](https://pure.uva.nl/ws/files/24636645/74286_312955.pdf) In 1925 the ISI meeting in Rome adopted a resolution accepting both randomization and purposive sampling.[20](https://www150.statcan.gc.ca/n1/pub/12-001-x/2013002/article/11883-eng.htm)
Neyman's 1934 paper to the [Royal Statistical Society](https://www.edgepedia.org/royal-statistical-society) divided the representative method into the method of stratified sampling and the method of purposive selection, the latter the procedure described by Bowley, Gini, and Galvani.[6](https://doi.org/10.2307/2342192) The paper presented the concept of confidence intervals and gave the theory for optimum allocation of sampling units to strata.[19](https://projecteuclid.org/journalArticle/Download?urlId=10.1214%2Fss%2F1177013352&isResultClick=False) His empirical evaluation of Italian census data showed that purposive sampling could produce unsatisfactory estimates, and the paper is widely recognized as establishing the merits of probability sampling.[17](https://pure.uva.nl/ws/files/24636645/74286_312955.pdf) • [5](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations) Kendall and Babington Smith's 1938 paper in the [Journal of the Royal Statistical Society](https://www.edgepedia.org/journal-of-the-royal-statistical-society) treated random sampling numbers and four tests of local randomness.[21](https://doi.org/10.2307/2980655) The classical theory was completed by Hansen and Hurwitz's 1943 multi-stage sampling theory with probability-proportional-to-size first-stage selection,[22](https://doi.org/10.1214/aoms/1177731356) Horvitz and Thompson's 1952 generalization of sampling without replacement,[14](https://doi.org/10.1080/01621459.1952.10483446) and the 1953 textbook of Hansen, Hurwitz and Madow.
## Variants
[Systematic sampling](https://www.edgepedia.org/systematic-sampling) selects every Kth unit after a random start between 1 and K; with a randomly ordered population it yields results similar to SRS, but a periodic feature in the list that coincides with the sampling interval can make samples unrepresentative.[23](https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm) • [24](https://theorempath.com/topics/survey-sampling-methods) Although only the start is random, it is a probability sampling method with known inclusion probabilities of n/N for every unit when the start is random.[25](https://uk.sagepub.com/sites/default/files/upm-assets/40803_book_item_40803.pdf) • [23](https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm) • [25](https://uk.sagepub.com/sites/default/files/upm-assets/40803_book_item_40803.pdf) [Stratified sampling](https://www.edgepedia.org/stratified-sampling) divides the population into strata and draws partial samples from each;[6](https://doi.org/10.2307/2342192) it improves variance over SRS, more so when strata differ greatly,[26](https://ar5iv.labs.arxiv.org/html/2011.00901) and, unlike quota sampling, it is a probability procedure that permits estimation of sampling error.[25](https://uk.sagepub.com/sites/default/files/upm-assets/40803_book_item_40803.pdf) [Cluster sampling](https://www.edgepedia.org/cluster-sampling) is in most cases less efficient than SRS because neighboring units tend to be alike; with equal cluster sizes B and intraclass correlation \( \rho \) the design effect is approximately \( \text{DEFF} \approx 1 + (B-1)\rho \).[23](https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm) • [24](https://theorempath.com/topics/survey-sampling-methods) Probability-proportional-to-size selection uses size information so larger units have a higher selection chance.[23](https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm) Quota sampling uses availability sampling and requires no frame; after the 1948 Gallup poll, which used quota sampling and incorrectly predicted 49.5% for Dewey against 44.5% for Truman, polling organizations shifted toward probability sampling.[25](https://uk.sagepub.com/sites/default/files/upm-assets/40803_book_item_40803.pdf) • [12](https://online.stat.psu.edu/stat506/Lesson01) Sudman's 1966 probability sampling with quotas was an early hybrid approach.[27](https://doi.org/10.1080/01621459.1966.10480903)
## Applications
For a SRSWOR mean the margin of error is \( \text{MOE} = Z\sqrt{(1-n/N)S_{y}^{2}/n} \).[4](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html) An unadjusted size \( n' = (Z/m)^{2} S_{Y}^{2} \) for desired margin m is corrected to \( n = n'/(1+n'/N) \).[4](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html) For a proportion with known N, \( n = z^{2} \cdot p \cdot q / [\text{ME}^{2} + z^{2} \cdot p \cdot q/N] \).[28](https://stattrek.org/sample-size/simple-random-sample?tutorial=samp) The fpc can be ignored when the sampling fraction is below 5% (up to 10% for many purposes); ignoring it overestimates the variance.[11](https://home.iitk.ac.in/~shalab/sampling/chapter2-sampling-simple-random-sampling.pdf) If nonresponse is expected at rate φ, select \( n_{\text{select}} = n/(1-\phi) \) units to achieve n responses.[4](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html)
## Limitations and alternatives
SRS requires no frame information beyond a complete list of units, but creating such a list for a large population can be expensive or unrealistic, and the design ignores auxiliary information that stratified sampling would exploit.[23](https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm) Nonresponse is a central failure mode: a flawless SRS of 1,000 people with a 20% response rate is no longer a random sample in the respondents unless nonresponse is shown to be unrelated to the study variables, and a large sample does not help if a large proportion fails to respond.[15](https://www.casrai.org/guides/simple-random-sampling) • [3](https://www.amsi.org.au/ESA_Senior_Years/PDF/RandomSampling4b.pdf) Undercoverage and nonresponse can represent specific subgroups and bias estimates.[29](https://eprints.soton.ac.uk/435300/1/WP5_Random_probability_vs_quota_sampling.pdf) Compensation inflates the base weights of similar responding elements for unit nonresponse and applies calibration adjustments such as post-stratification for noncoverage.[5](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations) Non-sampling error, from frame flaws, measurement error, and nonresponse bias, does not shrink with sample size and is not captured by a standard confidence interval.[15](https://www.casrai.org/guides/simple-random-sampling) Computer and calculator random numbers are pseudo-random, generated by a formula so the sequence is predetermined.[7](https://www.stat.auckland.ac.nz/~wild/ChanceEnc/Ch01.random.pdf) Haphazard and representative sampling do not yield simple random samples,[1](https://www.webpages.uidaho.edu/population_ecology/Simple_random_sampling.htm) and in arbitrary sampling the selection probabilities are unknown, so design-based estimation is impossible.[10](https://dickbrus.github.io/SpatialSamplingwithR/IntroProbabilitySampling.html) Without a probability selection mechanism there is no defined basis for computing a standard error or margin of error.[15](https://www.casrai.org/guides/simple-random-sampling) Even so, Brick has argued that a well-conducted probability sample with a low response rate is likely to be less biased than a volunteer sample, while declining response rates and the loss of effective frames are making probability sampling harder.[29](https://eprints.soton.ac.uk/435300/1/WP5_Random_probability_vs_quota_sampling.pdf)
## References
1. [Simple Random Sampling (Wlf 543, E. O. Garton, University of Idaho)](https://www.webpages.uidaho.edu/population_ecology/Simple_random_sampling.htm)
2. [Fundamentals of Applied Sampling (Piazza, UC Berkeley)](https://sda.berkeley.edu/dlab/sampling/Piazza_draft.pdf)
3. [Random sampling (AMSI Schools module)](https://www.amsi.org.au/ESA_Senior_Years/PDF/RandomSampling4b.pdf)
4. [Chapter 6 Simple Random Sampling, STAT392 Sample Surveys (Victoria University of Wellington)](https://homepages.ecs.vuw.ac.nz/~rarnold/STAT392/SampleSurveysBook/_book/simple-random-sampling.html)
5. [Sampling from finite populations (Encyclopedia of Mathematics)](https://encyclopediaofmath.org/wiki/Sampling_from_finite_populations)
6. [Jerzy Neyman (1934). On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. Journal Of The Royal Statistical Society.](https://doi.org/10.2307/2342192)
7. [Random Numbers (Chance Encounters, University of Auckland)](https://www.stat.auckland.ac.nz/~wild/ChanceEnc/Ch01.random.pdf)
8. [R: Random Samples and Permutations (sample / sample.int documentation)](https://stat.ethz.ch/R-manual/R-devel/library/base/html/sample.html)
9. [A review of model-assisted and design-based survey sampling (Tillé)](https://arxiv.org/pdf/1612.04965)
10. [Spatial sampling with R, Introduction to probability sampling (D. Brus)](https://dickbrus.github.io/SpatialSamplingwithR/IntroProbabilitySampling.html)
11. [Chapter 2: Simple Random Sampling (Shalabh, IIT Kanpur)](https://home.iitk.ac.in/~shalab/sampling/chapter2-sampling-simple-random-sampling.pdf)
12. [Estimating Population Mean and Total under SRS – STAT 506, Penn State](https://online.stat.psu.edu/stat506/Lesson01)
13. [Simple Random Sampling (Moulinath Banerjee, University of Michigan, 2012)](https://dept.stat.lsa.umich.edu/~moulib/sampling.pdf)
14. [D. G. Horvitz, D. J. Thompson (1952). A Generalization of Sampling Without Replacement from a Finite Universe. Journal of the American Statistical Association.](https://doi.org/10.1080/01621459.1952.10483446)
15. [Simple Random Sampling: Definition, Steps, and Worked Examples (CASRAI)](https://www.casrai.org/guides/simple-random-sampling)
16. [1.2.2 Sampling Methods (Penn State STAT 200)](https://online.stat.psu.edu/stat200/book/export/html/23)
17. [The rise of survey sampling (Bethlehem, University of Amsterdam)](https://pure.uva.nl/ws/files/24636645/74286_312955.pdf)
18. [Kiaer, Den repræsentative undersøgelsesmethode (1897, republished with English translation 1976)](https://www.ssb.no/a/histstat/sos/sos_027.pdf)
19. [History of survey sampling review (Statistical Science)](https://projecteuclid.org/journalArticle/Download?urlId=10.1214%2Fss%2F1177013352&isResultClick=False)
20. [Brewer, Three controversies in the history of survey sampling (Survey Methodology, 2013)](https://www150.statcan.gc.ca/n1/pub/12-001-x/2013002/article/11883-eng.htm)
21. [M. G. Kendall, B. Babington Smith (1938). Randomness and Random Sampling Numbers. Journal Of The Royal Statistical Society.](https://doi.org/10.2307/2980655)
22. [Morris H. Hansen, William N. Hurwitz (1943). On the Theory of Sampling from Finite Populations. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177731356)
23. [3.2.2 Probability sampling (Statistics Canada)](https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm)
24. [Survey Sampling Methods: SRS, Stratified, Cluster, and Multi-Stage Designs](https://theorempath.com/topics/survey-sampling-methods)
25. [Chapter 5: Choosing the Type of Probability Sampling (SAGE textbook chapter)](https://uk.sagepub.com/sites/default/files/upm-assets/40803_book_item_40803.pdf)
26. [Sampling Algorithms, from Survey Sampling to Monte Carlo Methods: Tutorial and Literature Review](https://ar5iv.labs.arxiv.org/html/2011.00901)
27. [Seymour Sudman (1966). Probability Sampling with Quotas. Journal of the American Statistical Association.](https://doi.org/10.1080/01621459.1966.10480903)
28. [Sample Size: Simple Random Samples (Stat Trek)](https://stattrek.org/sample-size/simple-random-sample?tutorial=samp)
29. [Random probability vs quota sampling (University of Southampton working paper)](https://eprints.soton.ac.uk/435300/1/WP5_Random_probability_vs_quota_sampling.pdf)
---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Sampling design and survey methodology › Sampling designs and estimators › Simple random sampling*
*Initially written Sep 29, 2026 · Edited: Oct 15, 2026; Oct 21, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license