Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Poisson sampling

Survey methodology process

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated September 23, 2025

In survey methodology, Poisson sampling (sometimes denoted as PO sampling) is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample.

Each element of the population may have a different probability of being included in the sample (\pi _{i}). The probability of being included in a sample during the drawing of a single sample is denoted as the first-order inclusion probability of that element (p_{i}). If all first-order inclusion probabilities are equal, Poisson sampling becomes equivalent to Bernoulli sampling, which can therefore be considered to be a special case of Poisson sampling.

The name conveys that the number of samples leads to a Poisson binomial distribution, which can approximate the Poisson distribution (via Le Cam's theorem).

01A mathematical consequence of Poisson sampling

Mathematically, the first-order inclusion probability of the ith element of the population is denoted by the symbol \pi _{i} and the second-order inclusion probability that a pair consisting of the ith and jth element of the population that is sampled is included in a sample during the drawing of a single sample is denoted by \pi _{ij}.

The following relation is valid during Poisson sampling when i\neq j (i.e., Independence):

\pi _{ij}=\pi _{i}\times \pi _{j}.

\pi _{ii} is defined to be \pi _{i}.

Watch videos about Poisson samplingExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Poisson sampling”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Bernoulli sampling

In the theory of finite population sampling, Bernoulli sampling is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample. An essential property of Bernoulli sampling is that all elements of the population have equal probability of being included in the sample.

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume).

Poisson point process

In probability theory, statistics and related fields, a Poisson point process is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution.