Skip to content

Bernoulli sampling

Select each eligible unit independently with the same preset inclusion probability, allowing the sample size to vary.

Version
v1 · 2026-10-17 · History
Domain-specific #
14066
Domain group
Formal Sciences
Origin domain
Experimental Design & Statistics
Subdomains
Probability Sampling, Survey Design → Experimental Design & Statistics
Aliases
Bernoulli sample selection

Core Idea

Bernoulli sampling is a selection design: each eligible unit independently receives the same preset probability \(p\) of entering a sample. It does not fix the number of units selected in advance. For a finite frame of \(N\) units under a pure common-\(p\) draw, the sample size is binomial, \(n_s\sim\operatorname{Binomial}(N,p)\), with expectation \(Np\) and variance \(Np(1-p)\).[1] The membership decision is the binary trial; a binary attribute measured on a unit after selection is a different variable.

The rule can act on business-survey units or database rows without changing its independent, equal-probability structure. Statistics Canada's Integrated Business Statistics Program names Bernoulli selection as its default method, with other design steps around it.[2] PostgreSQL offers TABLESAMPLE BERNOULLI to select or ignore individual rows independently at the specified percentage.[3] Neither application's broader purpose is a requirement of every Bernoulli draw.

Structural Signature

  • Eligible unit frame — a set of distinct units defines the candidates for inclusion in the finite-population formulation.[1] Without a specified frame, an inclusion probability has no identifiable subject.
  • Common probability — one \(p\) applies to every unit in the particular Bernoulli draw.[1] Different \(p_i\) values belong to a broader independent-inclusion design; a stratified program may use different common values in different strata.[2]
  • Independent binary trial — each unit is selected or rejected independently of the decisions for other units.[1] PostgreSQL's different SYSTEM method selects storage blocks.[3] Choosing exactly a target count or selecting a whole block changes the joint unit-inclusion structure.
  • Realized selected set — the positive trials form the sample; its count is an outcome, not a fixed input under the pure design.[1] Statistics Canada uses fixed-size SRS if a stratum draw falls below its threshold.[2] That fallback must be described separately.

What It Is Not

  • Not a Bernoulli-distributed study response. The sampling trial decides whether a unit is included; the unit's measured answer may have a different distribution.
  • Not fixed-size simple random sampling. A pure Bernoulli draw can return fewer or more than its expected \(Np\); fixing size couples the inclusion decisions.[1]
  • Not general unequal-probability Poisson sampling. In the cited survey-methods treatment, Bernoulli sampling is its equal-probability special case.[1]
  • Not PostgreSQL TABLESAMPLE SYSTEM. That method chooses storage blocks, then returns all rows in selected blocks; row memberships are not independent in the same way.[3]
  • Not guaranteed representativeness. Equal chance and independence specify a draw law, not completeness of the frame or similarity of a realized small sample to every population feature.

Scope of Application

Within survey design, the rule can be used inside a defined population or stratum, including business surveys.[2] Within data systems, it can sample individual table rows.[3] In both, the objects differ but the eligible-unit, common-\(p\), independent-decision and selected-set roles stay literal. A database command that samples blocks instead, or a survey that replaces a too-small draw with fixed-size SRS, has crossed the boundary for that component. These implementation qualifications matter when one reports the probability law for the final sample.

Clarity

The phrase “random sample” hides different joint laws. A fixed-size simple random sample, a random-start systematic sample, and a Bernoulli sample can give each unit equal marginal probability while differing in whether memberships are independent and whether the count is fixed. Bernoulli's test is therefore two-part: common inclusion probability and independent unit-level decisions. The title refers to this selection operation rather than the distribution of a recorded survey answer or the binomial count alone.

Manages Complexity

Instead of listing probabilities for every possible subset, the design can be specified by the frame and one \(p\): each membership indicator is an independent Bernoulli variable. The probability of a particular selected set of size \(n_s\) is \(p^{n_s}(1-p)^{N-n_s}\), and summing over sets of a given size produces the binomial sample-size law.[1] This compact description is valid for the pure common-\(p\) draw. It cannot be pasted onto the entire Statistics Canada program, which includes strata, certainty units and a low-count fixed-size fallback.[2]

Abstract Reasoning

The pure design has binomial sample size with expectation \(Np\).[1] That expected count is a planning value, not a guarantee; the chance of a small or empty draw can be computed before choosing \(p\). PostgreSQL's SYSTEM method selects blocks while BERNOULLI selects rows independently.[3] A block draw cannot be assumed to give independent row membership, so the analyst must inspect the actual selection operation. Estimation quality then requires a separate question about frame coverage, weighting and the quantity to be estimated.

Knowledge Transfer

Move from a business register to a SQL table: a business or a row becomes the unit, but the same independent common-\(p\) decision remains. That is literal transfer of a statistical selection design across carriers, not a metaphor. PostgreSQL scans the whole table for its Bernoulli method.[3] Statistics Canada's program uses a business frame and sampling-coordination rules.[2] Neither implementation choice changes the Bernoulli inclusion relation itself. Outside selection of a defined unit frame, a resemblance to “coin flips” is too broad to identify Bernoulli sampling. The wider idea of selection has other catalog identities.

Examples

Canonical: a Bernoulli step in Statistics Canada's business surveys

Statistics Canada describes Bernoulli sampling as the default selection method for its Integrated Business Statistics Program. The program first organizes business units into strata with assigned inclusion probabilities; some must-take units have probability one. Within an ordinary Bernoulli component, the eligible businesses receive the component's specified selection chance. If too few units are selected in a stratum, the agency uses a simple random sample instead.[2] The entire program is thus a composite design, and its final sample count should not be labeled one global \(\operatorname{Binomial}(N,p)\) draw.

Mapped back: business units in one eligible stratum → that component's common \(p\) → independent Bernoulli membership decisions → a realized count that may trigger a separate fallback. The fallback is a boundary condition, not part of the pure Bernoulli rule.

Applied: PostgreSQL row sampling

In PostgreSQL, TABLESAMPLE BERNOULLI (10) asks for independent inclusion of each scanned table row at a ten-percent probability. The returned row count is approximate, because the command does not promise exactly one tenth of the table. PostgreSQL contrasts this with TABLESAMPLE SYSTEM, which samples physical blocks and returns the rows they contain.[3] This is a database query operation, not a business-survey estimator.

Mapped back: table rows → \(p=0.10\) → independent row decisions during the scan → the query's variable-size selected subset. If the command used SYSTEM, the unit of randomization would be a block and the Bernoulli row-level mapping would fail.

Structural Tensions

Independent draw versus minimum stratum count. Keeping unconditioned unit decisions preserves the simple common-\(p\) law, but can yield too few surveyed businesses. Statistics Canada addresses that practical risk by substituting a simple random sample when a stratum falls below its threshold.[2] The substitution can secure the desired count, but the final selection is no longer solely the original Bernoulli draw. Diagnostic: Is a minimum count required for this stratum, and which actual design selected the final units?

Independent rows versus faster block selection. PostgreSQL's BERNOULLI method scans the whole table to decide about each row. Its SYSTEM method can be faster at small requested percentages by sampling blocks, but clustered records within a selected block travel together and may yield a less random row-level sample.[3] Diagnostic: Does the analysis require independently sampled rows strongly enough to pay for a full scan, or is block-level selection adequate?

Structural–Framed Character

Bernoulli sampling is strongly structural within its statistical domain. Its common probability and independent binary decisions can be stated without a particular institution, product or subject matter. Its evaluative weight is modest: the name gives a precise selection law, not a judgment that the realized sample is good enough for an inference. Its human-practice dependence lies in defining the eligible frame, choosing \(p\), implementing the random mechanism and deciding what to do with a small draw. Its institutional origin is survey and sampling methodology, while PostgreSQL demonstrates a computational implementation. Its vocabulary travel is high across tasks that literally select units, but the words frame, inclusion probability and sample are still statistical. The import-versus-recognition test is whether the new setting actually uses independent same-\(p\) membership decisions, rather than merely borrowing “Bernoulli” for a binary outcome. Its character: an exact selection structure whose named operative rule remains a probability-sampling method, not a universal principle of every kind of choice.

Structural Core vs. Domain Accent

The skeletal relation is independent retention of each eligible unit under a common probability. A Probability Measure is a strict prerequisite: its normalized product law on binary inclusion outcomes makes the draw and sample-size probabilities coherent, but the draw is a procedure rather than a kind of measure. Selection is a broader catalog pattern about differential retention from available alternatives, and Sampling (Representativeness) concerns the relation of subset selection to a target population and inference; neither automatically supplies Bernoulli's joint probability law. The named entry's domain accent is not incidental notation: eligibility, inclusion probability, realized sample and the binomial count are all parts of statistical sampling. Business units and database rows are unlike carriers but share that exact domain operation. An office hiring process might flip coins independently, yet without a defined statistical sample it would be analogy or a new selection use, not evidence that the named method is a substrate-independent Prime.

This entry presupposes Probability measure.

Bernoulli sampling depends, in every case, on a Probability Measure: the finite product law over inclusion outcomes requires one. Probability Measure in turn falls under Probability, and Probability under Measure, so Bernoulli sampling relates to those through it rather than directly. Sampling (Representativeness) is a broader concept with aliases including probability sampling, survey sampling and sample selection. Bernoulli sampling can provide a known-probability selection step for such an inferential design, but a PostgreSQL row subset need not be used for population estimation or be representative. Selection concerns differential retention and composition change; equal ex ante inclusion chances, including possible all-in or none-in outcomes, do not make Bernoulli sampling a kind of Selection in every case. Neither neighbor's name alone establishes the full Bernoulli design.

Relationships to Other Abstractions

Local relationship map for Bernoulli samplingParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Bernoulli samplingDOMAINDomain-specific abstraction: Probability measure — presupposesProbabilitymeasureDOMAIN

Current abstraction Bernoulli sampling Domain-specific

Parents (1) — more general patterns this builds on

  • Bernoulli sampling presupposes Probability measure Domain-specific

    Bernoulli sampling requires a normalized product probability measure over unit-inclusion outcomes.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Bernoulli sampling sits in a sparse region of the domain-specific corpus (97th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2721 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-17

Not to Be Confused With

  • Systematic Sampling: a random start and regular interval in an ordered frame; selections are not independent unit flips.
  • Correct Sampling: Gy's particulate-material condition of equal nonzero particle inclusion opportunity; equality alone does not supply Bernoulli independence.
  • Multistage Sampling: a design that selects through nested stages, not one common-\(p\) trial over the final units.
  • Sampling Frame: the list or specification of eligible units; Bernoulli sampling is an operation on such a frame.
  • Binomial sample-size distribution: a consequence of the pure finite design, not the selection rule itself.[1]

References

[1] Ismo Teikari, Poisson Mixture Sampling in Controlling the Distribution of Response Burden in Longitudinal and Cross Section Business Surveys, Statistics Finland Research Reports 232 (2001), printed pp. 23–26, PDF pp. 24–27. https://otos.stat.fi/bitstreams/950353df-68a8-4fd6-bda6-05b207e094f5/download . Full Statistics Finland PDF inspected; Bernoulli membership indicators, binomial count and equal-probability relation to Poisson sampling are stated there. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j

[2] Statistics Canada, “An Overview of the Integrated Business Statistics Program Sampling Methodology,” especially “Sample selection and coordination,” “Allocation,” and “Must-Take Units.” https://www.statcan.gc.ca/en/statistical-programs/document/IBSP_D2_T9_V1 . Full official page inspected; the program includes stratification and non-Bernoulli components. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[3] PostgreSQL Global Development Group, PostgreSQL 10 Documentation, “SELECT,” FROM clause TABLESAMPLE paragraphs. https://www.postgresql.org/docs/10/sql-select.html . Full official documentation inspected; BERNOULLI operates on individual rows and SYSTEM on blocks. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h