Home Statistical Dictionary Probability Proportional to Size Sampling

Probability Proportional to Size Sampling

A probability sampling design, common at the first stage of multistage surveys, in which each unit's probability of selection is made proportional to a known measure of its size, so larger units are more likely to be chosen.

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In Plain English

Probability proportional to size (PPS) sampling deliberately tilts the odds so that bigger units, more populous cities, larger schools, higher-revenue firms, are more likely to be picked than smaller ones, using some known 'size' measure as the basis. This sounds like it would bias the results, but it's actually a clever efficiency trick for multistage designs: if you're eventually going to sample a fixed number of people within each selected cluster regardless of the cluster's size, giving big clusters a bigger chance of being chosen in the first place keeps each individual person's overall probability of ending up in the final sample roughly equal, which is exactly what you want.

Definition

In probability proportional to size (PPS) sampling, the probability of selecting unit $i$ is set proportional to a known auxiliary size measure $Z_{i}$ (such as population count, enrollment, or revenue), $π_{i} \propto Z_{i}$ , rather than being equal across units. PPS is most commonly used at the first stage of a multistage design, where the primary sampling units are clusters of unequal size, combining PPS selection of clusters with a fixed number of units sampled per selected cluster keeps the overall probability of selecting any individual final-stage unit approximately constant across clusters (a self-weighting design), which both simplifies estimation (weights are close to uniform) and improves precision relative to selecting clusters with equal probability, since it avoids either oversampling small clusters or leaving large clusters underrepresented relative to their share of the population.

Formula

π_{i} = \frac{m \cdot Z _{i}}{\sum _{j = 1}^{M} Z _{j}}

Notation

Z_{i}

—Known size measure for unit i (e.g. population, enrollment)

π_{i}

—Selection probability of unit i, proportional to Z_i

m

—Number of units selected from M total

Properties

PPS sampling is what makes 'take a fixed number of respondents per selected cluster, regardless of the cluster's actual size' a statistically sound strategy, without PPS weighting at the cluster-selection stage, this fixed-sample-per-cluster approach would systematically overrepresent people living in small clusters and underrepresent people in large ones.
A design that combines PPS selection of clusters with equal sample sizes within each selected cluster is called self-weighting, because every final-stage unit ends up with approximately the same overall selection probability, which greatly simplifies the analysis (survey weights are close to constant) compared to a design with widely varying weights.
PPS selection at the first stage is typically implemented using systematic sampling with a cumulative-size ordering (list clusters, compute cumulative size totals, then systematically sample positions along that cumulative scale), a practical algorithm that guarantees each cluster's selection probability is exactly proportional to its size measure.

At a Glance

Ruleselection probability is proportional to a known size measure

Typical usefirst-stage cluster selection in multistage designs

Benefitcombined with fixed within-cluster sample size, yields a self-weighting design

Requiresan auxiliary size variable known for every unit before sampling

Last updated September 8, 2026← Back to the Dictionary