Sample Size When the Population Size Is Unknown

Distinguish two different unknown quantities

An unknown population size means the total number of eligible units N is not available. An unknown population proportion means the fraction p with the outcome of interest is not known in advance. These are separate uncertainties. A large-population sample-size formula can omit N while still needing a planning value for p. It also needs a confidence level, a precision target and a suitable sampling method. Missing population size does not make every other design choice disappear.

Use the large-population approximation

For estimating one proportion by an approximate normal interval, the initial target is n₀=z²p(1−p)/e². There is no N term in this expression. If the eligible population is large compared with the planned sample, finite-population correction has little effect. When N is unknown, leave the optional population field blank instead of entering zero or a guessed small number. Treat the result as a planning calculation under the stated large-population assumption, not a count derived from knowledge you do not have.

Handle an unknown expected proportion

Within the simple formula, p(1−p) is largest at p=0.5. Using 50% therefore gives the largest target across possible proportions for fixed z and e. At 95% confidence and a five-percentage-point margin, the unrounded count is 384.16 and the form returns 385. This conservative choice concerns variance in the formula only. It does not protect against an unrepresentative sample, missing groups, inaccurate responses or a rare-event objective requiring a different precision definition.

Change precision and verify the consequence

Keeping 95% confidence and p=0.5 but narrowing the margin from five to three percentage points gives 1.96²×0.25/0.03²≈1067.1111, rounded up to 1068. Enter 3 for the margin to check the second result. The target grows as the inverse square of the margin: a three-point margin requires about (5/3)² times the unrounded count for five points. This growth explains why a modest-sounding improvement in precision can require substantially more observations.

Revisit the plan if a reliable population count appears

Suppose a complete, relevant population frame later establishes N=1000. Under simple random sampling without replacement, applying the correction to the original five-point example gives a target of 278 rather than 385. That reduction is justified only for that defined finite population and design. A count of currently reachable people is not automatically the total eligible population. If inference concerns a broader or changing population, the smaller frame count may not support the same correction.

Separate the frame from the target population

A target population describes all units about which a conclusion is intended. A sampling frame is the operational list or mechanism used to select observations. If some eligible units have no chance of selection, increasing the sample from the available frame does not restore their representation. Record coverage limitations and how selection probabilities arise. The calculation of 385 or 1068 presumes an appropriate sampling model; it does not validate an open volunteer survey or a convenience sample as though it were random sampling.

Account for subgroups and usable responses

A total sample that meets an overall precision target may leave small subgroups with very imprecise estimates. Plan important subgroups explicitly and consider stratification or allocation rather than assuming the overall count applies separately to each group. Likewise, if only 75% of invitations are expected to yield usable observations, divide the required complete count by 0.75 and round upward for an operational invitation target. These adjustments answer separate questions and should remain visible rather than being hidden inside a guessed population field.

Report the assumptions that replaced the missing N

State that the population count was unknown and the large-population formula was used, along with p, confidence, margin and the rounded target. Explain why the sampling method supports the intended inference and whether response or design adjustments were added. If the research question concerns a mean, a diagnostic accuracy parameter, a matched case-control comparison or statistical power, select its own planning method. The absence of N does not make the one-proportion formula universal across those objectives.

Related guides

Calculate Sample Size for a Population Proportion

Choose confidence, precision and an expected proportion, calculate two sample targets, apply a finite-population correction, and round upward.

Calculate Sample Size from a Confidence-Interval Target

Turn interval width into a margin, derive sample targets for proportions and means, and explain why a published interval may not reveal a unique count.

How Large Should a Sample Be for Useful Precision?

Compare precision and confidence targets, distinguish estimation from power, and plan subgroup, design and response needs instead of using a fixed rule.