CoChat
Sign Up
Cited by 3,606Survey Sampling and Estimation TechniquesCensus and Population EstimationSurvey Methodology and Nonresponse

Snowball Sampling

Open access
Goodman, Leo A.·1961·The Annals of Mathematical Statistics
DOI 10.1214/aoms/1177705148
Open this paper in CoChat — free accountOpen PDF

Abstract

An $s$ stage $k$ name snowball sampling procedure is defined as follows: A random sample of individuals is drawn from a given finite population. (The kind of random sample will be discussed later in this section.) Each individual in the sample is asked to name $k$ different individuals in the population, where $k$ is a specified integer; for example, each individual may be asked to name his "$k$ best friends," or the "$k$ individuals with whom he most frequently associates," or the "$k$ individuals whose opinions he most frequently seeks," etc. (For the sake of simplicity, we assume throughout that an individual cannot include himself in his list of $k$ individuals.) The individuals who were not in the random sample but were named by individuals in it form the first stage. Each of the individuals in the first stage is then asked to name $k$ different individuals. (We assume that the question asked of the individuals in the random sample and of those in each stage is the same and that $k$ is the same.) The individuals who were not in the random sample nor in the first stage but were named by individuals who were in the first stage form the second stage. Each of the individuals in the second stage is then asked to name $k$ different individuals. The individuals who were not in the random sample nor in the first or second stages but were named by individuals who were in the second stage form the third stage. Each of the individuals in the third stage is then asked to name $k$ different individuals. This procedure is continued until each of the individuals in the $s$th stage has been asked to name $k$ different individuals. The data obtained using an $s$ stage $k$ name snowball sampling procedure can be utilized to make statistical inferences about various aspects of the relationships present in the population. The relationships present, in the hypothetical situation where each individual in the population is asked to name $k$ different individuals, can be described by a matrix with rows and columns corresponding to the members of the population, rows for the individuals naming and columns for the individuals named, where the entry $\theta_{ij}$ in the $i$th row and $j$th column is 1 if the $i$th individual in the population includes the $j$th individual among the $k$ individuals he would name, and it is 0 otherwise. While the matrix of the $\theta$'s cannot be known in general unless every individual in the population is interviewed (i.e., asked to name $k$ different individuals), it will be possible to make statistical inferences about various aspects of this matrix from the data obtained using an $s$ stage $k$ name snowball sampling procedure. For example, when $s = k = 1$, the number, $M_{11}$, of mutual relationships present in the population (i.e., the number of values $i$ with $\theta_{ij} = \theta_{ji} = 1$ for some value of $j > i$) can be estimated. The methods of statistical inference applied to the data obtained from an $s$ stage $k$ name snowball sample will of course depend on the kind of random sample drawn as the initial step. In most of the present paper, we shall suppose that a random sample (i.e., the "zero stage" in snowball sample) is drawn so that the probability, $p$, that a given individual in the population will be in the sample is independent of whether a different given individual has appeared. This kind of sampling has been called binomial sampling; the specified value of $p$ (assumed known) has been called the sampling fraction [4]. This sampling scheme might also be described by saying that a given individual is included in the sample just when a coin, which has a probability $p$ of "heads," comes up "heads," where the tosses of the coin from individual to individual are independent. (To each individual there corresponds an independent Bernoulli trial determining whether he will or will not be included in the sample.) This sampling scheme differs in some respects from the more usual models where the sample size is fixed in advance or where the ratio of the sample size to the population size (i.e., the sample size-population size ratio) is fixed. For binomial sampling, this ratio is a random variable whose expected value is $p$. (The variance of this ratio approaches zero as the population becomes infinite.) In some situations (where, for example, the variance of this ratio is near zero), mathematical results obtained for binomial sampling are sometimes quite similar to results obtained using some of the more usual sampling models (see [4], [7]; compare the variance formulas in [3] and [5]); in such cases it will often not make much difference, from a practical point of view, which sampling model is utilized. (In Section 6 of the present paper some results for snowball sampling based on an initial sample of the more usual kind are obtained and compared with results presented in the earlier sections of this paper obtained for snowball sampling based on an initial binomial sample.) Fo

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1214/aoms/1177705148

Cites

Statistics

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The review used chain-referral sampling from 20 core articles until saturation was reached.

    “These 20 core articles (Appendix II) served as “seeds” for a chain-referral sampling approach (Figure 1), also known as snowball sampling (Biernacki & Waldorf, 1981;Goodman, 1961).”

    From The Potential of Exam Wrappers in Higher Education Assessment Practice: Fostering Self-Efficacy Through Guided Self-Reflection (2025) · cites this paper as Goodman, 1961

  • Snowball sampling introduces potential selection bias and may limit the representativeness of participants' perspectives.

    “although snowball sampling facilitated access to professional screenwriters, it introduces potential selection bias and may limit the representativeness of participants' perspectives [21].”

    From How Do Human Creators Embrace Human-AI Co-Creation? A Perspective on Human Agency of Screenwriters (2026) · cites this paper as [21]

    The Introduction (p. 1) defines snowball sampling, and the summary (p. 2) discusses an initial random sample and unbiased estimators. Neither those passages nor searches for selection bias and representativeness address whether participants’ perspectives may be unrepresentative. (LLM, conf=0.75)

    Experimental — accuracy not yet measured

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Cites

No references could be resolved yet.

Research works printed but not linked: 8

  1. [1][1] DAvID BLAackwELL, "Conditional expectation and unbiased sequential estimation," Ann. Math. Stat., Vol. 18 (1947), pp. 105–110. -
  2. [2][2] JAmEs S. CoLEmAN, "Relational analysis: The study of social organizations with survey methods,"Human Organization, Vol. 17 (1958–59), pp. 28–36. -
  3. [3][3] W. EDwARDs DEMING AND GERALD J. GLAssER, "On the problem of matching lists by samples," J. Amer. Stat. Assn., Vol. 54 (1959), pp. 403–415. -
  4. [4][4] LEo A. GoodmAN, "On the estimation of the number of classes in a population,"Ann. Math. Stat., Vol. 20 (1949), pp. 572–579. -
  5. [5][5] LEo A. GoodmAN, "On the analysis of samples from k lists,"Ann. Math. Stat., Vol. 23 (1952), pp. 632–639. -
  6. [6][6] J. L. HodGEs, JR. AND E. L. LEHmANN, "Some problems in minimax point estimation," Ann. Math. Stat., Vol. 21 (1950), pp. 182–197. -
  7. [7][7] FREDERIcK MosTELLER, "Questions and Answers,"Amer. Statistician, Vol. 3 (1949), No. 3, pp. 12-13. -
  8. [8][8] MartıN A. Trow, "Right wing radicalism and political intolerance: A study of support for McCarthy in a New England town," Ph.D. dissertation, Columbia University, 1957.

Cited by

Sorted by citations

Related

Authors
Goodman, L. A.
Corresponding author
Goodman, Leo A.
Venue
The Annals of Mathematical Statistics
Publisher
Institute of Mathematical Statistics
Published
1961-03
Volume
32
Issue
1
Pages
148-170
ISSN
0003-4851 (ISSN-L)
DOI
10.1214/aoms/1177705148
OpenAlex
W4241491750
Cited by
3,606
References
0
Citations by year
2026: 297 · 2025: 364 · 2024: 310 · 2023: 280 · 2022: 330 · 2021: 299 · 2020: 260 · 2019: 194 · 2018: 168 · 2017: 176 · 2016: 162 · 2015: 134 · 2014: 118 · 2013: 88 · 2012: 79
FWCI
0.00
Type
journal article
Language
en
Open access
Locations
  • The Annals of Mathematical Statistics (publishedVersion)
  • renati (submittedVersion, other-oa)
  • Project Euclid (Cornell University) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Statistics and Probability › Survey Sampling and Estimation Techniques
  • Physical Sciences › Mathematics › Statistics and Probability › Census and Population Estimation
  • Social Sciences › Sociology and Political Science › Survey Methodology and Nonresponse
Keywords
snowball sampling, social networks, network relationships, finite population, sampling design

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review