Does PPS give every unit the same chance?

No. Chances depend on the size measure and the sampling design.

Must a sampled unit be selected only once?

No. With-replacement PPS allows repeats; without-replacement designs do not.

Does a larger size measure guarantee selection?

Not generally. Some without-replacement designs designate sufficiently large units as certainty selections, but that is a specific design feature.

AI Glossary / Statistics and probability / Intermediate

Probability proportional to size

Definition

Probability proportional to size, often abbreviated PPS, describes sampling designs in which selection probabilities are linked proportionally to a known size measure for each population unit. Larger units receive greater selection chances. The size measure might be employment, household count, or another relevant auxiliary quantity. The exact probability depends on the design: a per-draw probability in sampling with replacement is different from a unit's probability of appearing at least once in the final sample.

What it means

PPS allows a survey to devote more sampling attention to units that contribute more to a total of interest. It can improve precision when the size measure is informative about the quantity being estimated. It does not simply mean choosing the largest units, because selection remains probabilistic.

The unequal chances must be reflected in estimation. Otherwise, a sample deliberately favoring large units may be mistaken for an equally representative collection of units. The sampling frame should contain appropriate size information for every eligible unit, including a plan for units with zero, missing, or outdated measures.

How it works

For independent draws with replacement, assign unit i a positive size s_i and calculate its per-draw probability p_i = s_i divided by the sum of all size measures. Draw using these probabilities, allowing the same unit to be selected more than once.

For a total, one unbiased estimator in this design averages y_i/p_i across the draws, retaining repeated selections. Its expectation equals the population total because each contribution is weighted by its own draw probability. This is the Hansen - Hurwitz approach.

Without-replacement PPS designs require different selection and estimation details. Inclusion probabilities must be valid and cannot exceed one; very large units may need certainty selection. Do not take a with-replacement formula and apply it unchanged to another design. Variance estimation also needs the actual sampling structure, including clustering and any additional stages of selection.

Three practical examples

A business survey gives larger employers higher selection chances when estimating a total associated with employment.
A household survey selects geographic clusters using population counts before sampling households within selected clusters.
An operations audit samples service centers using workload as a size measure when estimating total processing effort.

Think of it this way

Think of a draw in which each unit receives a length of tape proportional to its size. A uniformly chosen position on the complete tape selects the unit covering that position. Larger sections are more likely to be selected, but smaller sections still have a known chance.

Worked example

In an original example, four service centers have size measures 10, 20, 30, and 40. Their per-draw probabilities under PPS with replacement are 0.10, 0.20, 0.30, and 0.40.

Suppose their actual totals of interest are 12, 24, 27, and 44, so the population total is 107. Three independent draws select the second, fourth, and fourth centers. The estimated total is:

(24/0.20 + 44/0.40 + 44/0.40)/3 = (120 + 110 + 110)/3 = 113.33 approximately.

It differs from 107 because a particular sample need not equal its expectation. Across repeated sampling, the expected single-draw contribution is 0.10 × 120 + 0.20 × 120 + 0.30 × 90 + 0.40 × 110 = 107.

The fourth center's probability of appearing at least once in three draws is 1 − 0.60³ = 0.784, not its per-draw probability of 0.40. The distinction determines which estimator and weights are appropriate.

Illustrative case study

Fictional case study: Cedar Operations plans a review of workload across service centers. Centers differ substantially in size, and equal selection probabilities would sometimes omit most high-volume operations.

The survey team proposes PPS using a documented workload measure. It records the exact selection method and retains the probabilities needed for estimation. Analysts review whether last year's workload is still informative and account for repeated selections or additional sampling stages. This fictional example illustrates survey design choices, not a claim that PPS always improves precision.

Common misconceptions and mistakes

A common mistake is treating PPS as selection of only the largest units. Another is analyzing the selected observations as an unweighted simple random sample. Both can undermine the intended population inference.

Do not confuse the measure of size with the variable being estimated; they may be related without being identical. A poor size measure can reduce the expected efficiency advantage. Also distinguish per-draw probabilities from final inclusion probabilities. Removing duplicate draws and keeping the original with-replacement estimator changes the procedure. Finally, PPS does not fix nonresponse, incomplete coverage, or inaccurate measurements merely by using known selection probabilities.

Business use

PPS can make surveys and audits more informative when units differ substantially in their contribution to business totals. It is especially useful when suitable auxiliary size information already exists across the sampling frame.

The practical cost is greater design and analysis complexity. Preserve the frame version, size measures, randomization procedure, selection probabilities, and treatment of certainty units. If AI helps process sampled records, retain this survey information through the data pipeline so that later summaries do not accidentally discard the design and its required weighting.

Check your understanding

Two units have size measures 15 and 45. What are their probabilities on one PPS draw with replacement?

Answer: Their probabilities are 15/60 = 0.25 and 45/60 = 0.75. These are draw probabilities; inclusion probabilities over several draws require a separate calculation.

Related terms

Horvitz - Thompson estimator
Cluster Sampling
Probability sampling
sampling frame
inclusion probability
sampling weight
auxiliary variable
Hansen–Hurwitz estimator
design-based inference

Learn next

Learn the difference between draws and included units, then study unequal-probability estimators. Practice PPS with replacement before comparing without-replacement designs, certainty selection, and multistage surveys. Include design-appropriate variance estimation and nonresponse adjustment when moving from examples to real surveys.

Variance estimation

Sources

Statistics Canada explains probability sampling and PPS: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch13/prob/5214899-eng.htm

A primary Survey Methodology article states the with-replacement PPS setup and per-draw probabilities: https://www150.statcan.gc.ca/n1/pub/12-001-x/2020002/article/00001/02-eng.htm

The four-center estimator calculation is original and uses ordinary independent PPS draws, not the article's additional ranked-set procedure. The fictional case is illustrative.

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