What Happens if the Eligible Population Is Smaller Than the Calculated Sample Size?

Your Calculation Requires 300 Participants, but Only 180 People Are Eligible. What Now?

You are planning a study of university teachers who have developed fully online courses. Your sample-size calculation indicates that 300 participants are required, but institutional records show that only 180 teachers meet the eligibility criteria.

Should you recruit all 180 and describe the study as a census? Apply a finite population correction to reduce the required sample? Expand recruitment to other institutions? Or proceed with fewer participants and acknowledge that the original statistical requirements cannot be met?

These options have different methodological implications. The appropriate response depends on the research question, target population, sampling design, and purpose of the original sample-size calculation.

A small eligible population is not automatically a research failure, but it requires researchers to distinguish what is mathematically possible from what is scientifically sufficient.

Reassess the Statistical Objective and Population Before Changing the Sample Size

Responding to a Sample-Size Requirement That Exceeds the Eligible Population

How Can the Required Sample Exceed the Population?

Many sample-size formulas are developed under assumptions appropriate to large populations or effectively unlimited sampling frames.

For example, a conventional calculation for estimating a population proportion with a specified confidence level and margin of error may initially produce a requirement larger than a small finite population.

In other circumstances, a power calculation may require more participants than exist because the expected effect is small, the outcome is highly variable, or the desired statistical power is demanding.

These situations should not be treated identically. A finite population adjustment may be relevant to some estimation problems but not to every hypothesis test or statistical model.

What Is the Difference Between the Target Population and the Eligible Population?

The target population is the group to which the research question and intended inference refer.

The eligible population consists of individuals satisfying the study's inclusion and exclusion criteria.

The accessible population is the subset researchers can legitimately identify and approach.

For example, a study may target online university teachers nationally but recruit only from one institution. The institution's 180 eligible teachers do not necessarily constitute the entire target population.

Researchers should clarify these distinctions before deciding whether the population is genuinely finite and whether a census is meaningful.

What Should You Verify First?

Before redesigning the study, check whether the sample-size calculation matches the actual research objective.

Confirm the primary outcome, statistical method, expected effect or variability, confidence level or power, and relevant sampling assumptions.

Also verify whether the population count is current and whether eligibility criteria have been applied correctly.

An incorrectly specified calculation or outdated population estimate may create an apparent shortfall that does not reflect the actual study requirements.

However, the calculation should not be manipulated retrospectively merely to produce a convenient number.

Can You Recruit the Entire Eligible Population?

Yes. Attempting to include every eligible member of a defined population is commonly described as a census approach.

For example, a researcher studying the 180 eligible online course designers within one institution may invite all 180.

If all participate and provide the required measurements, the researcher may obtain complete coverage of that defined population at the time of study.

However, inviting everyone does not mean everyone will participate.

The achieved dataset should not be described as a complete census if eligible individuals are missing because of refusal, noncontact, or incomplete data.

Does a Census Eliminate Sampling Error?

For descriptive characteristics of a fully observed, precisely defined finite population, a complete census eliminates uncertainty arising from selecting only a sample of that population.

However, it does not eliminate measurement error, nonresponse, data-processing errors, or other sources of bias.

Nor does it automatically eliminate uncertainty about future populations, other institutions, or a broader conceptual population.

For example, measuring every eligible teacher at one university does not establish how teachers at other universities would respond.

Researchers should distinguish finite-population description from broader statistical inference.

What Is the Finite Population Correction?

The finite population correction, or FPC, accounts for the reduction in sampling variance when a substantial proportion of a finite population is sampled without replacement under an appropriate probability-sampling design.

For simple random sampling without replacement, a commonly used correction to the standard error involves the following factor:

When the sample constitutes a small fraction of a large population, the correction is close to one. As the sampling fraction increases, the correction becomes more consequential.

The FPC is not a universal adjustment for nonprobability samples, randomized experiments, or all forms of model-based inference.

How Can the Finite Population Correction Reduce a Sample-Size Requirement?

For a conventional simple random sample intended to estimate a finite-population proportion, an approximate adjusted sample size can be calculated from an initial large-population requirement.

The result does not mean that 113 participants are sufficient for every research question involving 180 eligible people.

The adjustment is appropriate only when the original calculation, sampling design, and inferential target justify finite-population treatment.

When Is Finite Population Correction Inappropriate?

The correction should not be applied automatically when the sample-size requirement comes from statistical power for a randomized intervention, a regression model, or an analysis intended to generalize beyond the enumerated finite population.

For example, a study comparing two instructional interventions may require 300 participants to detect a particular effect with prespecified power.

The fact that only 180 eligible students exist does not ordinarily permit the researcher to reduce the power-based requirement to 113 through the finite population formula.

The study may remain underpowered for the original hypothesis.

Researchers should identify the statistical basis of the original calculation before considering any correction.

What Happens to Statistical Power When Fewer Participants Are Available?

When the achieved sample is smaller than the sample assumed in a power calculation, statistical power will often be lower, all else being equal.

The size of the reduction depends on the statistical test, effect size, outcome variability, allocation, and other design assumptions.

For example, a study designed for 80% power with 300 participants will not generally retain the same power with only 180 participants.

Researchers can calculate the power associated with the achievable sample under prespecified plausible effects, or determine the effect sizes that the study can detect with specified power.

Such calculations should be interpreted as planning assessments, not as post hoc explanations that a nonsignificant result proves the absence of an effect.

Should You Lower the Desired Statistical Power?

Lowering the target power reduces the calculated sample requirement, but it also increases the probability of failing to reject the null hypothesis when the specified alternative is true.

This may be acceptable in some exploratory or preliminary designs when the limitations are explicit.

However, reducing power solely to make the required sample fit the accessible population is not a substantive methodological justification.

Researchers should reconsider the scientific objective and whether a less definitive study would still provide useful evidence.

Any change to the original design should be documented transparently.

Can You Change the Expected Effect Size?

Increasing the assumed effect size often reduces the required sample in power-based calculations.

However, selecting an unrealistically large effect merely to obtain a feasible sample is inappropriate.

The effect-size assumption should be supported by relevant prior evidence, scientific reasoning, or a defensible smallest effect size of interest.

If the study cannot detect effects that would be scientifically important, this limitation should be acknowledged.

A convenient effect-size assumption does not resolve the underlying shortage of information.

Can Precision-Based Planning Offer an Alternative?

Sometimes. If the primary objective is estimation rather than hypothesis testing, researchers may consider the confidence interval width achievable with the available population.

For example, a study may seek to estimate the proportion of eligible teachers using a particular assessment practice.

With a finite population of 180, a probability sample of 120 may provide useful precision under appropriate assumptions, even when an initial large-population calculation suggested a larger sample.

However, the acceptable precision should be determined by the research question rather than chosen solely because it matches the available participants.

The distinction between precision-based and power-based sample-size planning is therefore important.

Can You Expand Recruitment to Other Institutions?

Yes, when the expanded recruitment setting remains consistent with the research question and target population.

For example, a study of online course designers across higher education institutions may reasonably include additional universities.

However, expanding beyond the original institution can introduce differences in policies, instructional practices, student populations, and other contextual characteristics.

The sampling strategy and analysis may need to account for site-level variation.

Additional institutional permissions and ethics approvals may also be required.

Expanding recruitment should strengthen alignment with the intended population rather than merely increase numbers.

Can Eligibility Criteria Be Broadened?

Possibly, but only when the revised criteria remain scientifically defensible.

Suppose a study originally includes only teachers with at least ten years of online teaching experience.

If that threshold is not essential to the research question, researchers might reconsider whether less experienced teachers also belong to the intended population.

However, broadening criteria can change the population and interpretation of the findings.

Researchers should not remove essential criteria merely to increase enrollment.

Material changes require appropriate protocol review and transparent reporting.

Can the Study Be Redesigned Around the Available Population?

Yes. A smaller eligible population may justify a different design if the revised study addresses a scientifically meaningful question.

For example, an underpowered comparative intervention study might be redesigned as a feasibility study examining implementation, acceptability, or recruitment procedures.

Alternatively, a broad survey may become a carefully bounded institutional case study.

Such changes alter the research contribution and should not be presented as equivalent to the original design.

The revised methodology must have its own appropriate sample justification.

What If the Entire Population Is Only 30 People?

A very small population may still support useful research, depending on the aim.

For example, a study examining the practices of all 30 members of a specialized academic committee could provide a detailed account of that particular group.

However, 30 participants may be insufficient for certain regression models, subgroup comparisons, or small-effect hypothesis tests.

The researcher should consider whether a census, qualitative approach, descriptive analysis, or alternative design best addresses the question.

The population's small size does not automatically make the research invalid, but it constrains the claims that can be supported.

Does Recruiting Everyone Solve Nonresponse Bias?

No. Inviting the entire eligible population does not guarantee complete participation.

If individuals who decline differ systematically from those who participate, the achieved data may be biased.

For example, teachers experiencing substantial workload difficulties may be less likely to complete a survey about instructional workload.

The resulting responses could underestimate the prevalence or severity of those difficulties.

Researchers should use ethically appropriate recruitment and follow-up procedures and report participation rates transparently.

What If the Eligible Population Is Smaller Than the Recruitment Contact Target but Not the Analytical Sample?

This is a different feasibility problem.

Suppose the study requires 150 usable participants and 250 individuals are eligible, but an anticipated 50% usable-data yield implies that 300 people would need to be approached.

The analytical target is smaller than the population, yet the expected recruitment process cannot achieve it from the available pool.

The researcher may need to improve recruitment yield, extend access, or reconsider the design.

The assessment of recruitment feasibility should therefore distinguish analytical requirements from expected recruitment losses.

How Should Uncertainty Be Reported With a Small Population?

Researchers should explain the population definition, sampling fraction, achieved participation, and inferential target.

Where a finite population correction is used, its assumptions and justification should be stated.

Where the study is underpowered or has limited precision, the implications should be discussed without treating nonsignificant findings as proof of no effect.

Confidence intervals may help communicate uncertainty, but their interpretation depends on the design and analysis.

Researchers should also describe possible selection, nonresponse, and measurement biases that numerical precision alone cannot capture.

How Should the Decision Be Documented in the Methodology?

The methods section should report the original sample-size requirement, the verified eligible population, and the reason the two differ.

It should then explain the chosen response, whether a census, appropriate finite population adjustment, additional sites, or a redesigned study.

If the final sample does not meet the original statistical requirements, researchers should state the resulting limitations.

Changes made after the original protocol should be reported and authorized where required.

A transparent explanation is preferable to presenting a revised convenient sample size as though it had always been the original methodological target.

Reassessing a Sample-Size Requirement of 300 With Only 180 Eligible Teachers

The example demonstrates a situation in which finite-population adjustment may legitimately change a precision-based requirement.

If the original 300 participants had instead been required for a powered intervention comparison, the same adjustment would not establish that 113 participants were sufficient.

Common Misconceptions About Small Eligible Populations

Match the Study's Claims to the Population and Information Actually Available

When the eligible population is smaller than the required sample, begin by confirming the population definition and the purpose of the original calculation.

Then identify which design adjustments are scientifically legitimate rather than choosing the smallest convenient number.

A small eligible population may support valuable research, but the study's conclusions must remain consistent with the information that population can provide.

Before Proceeding With a Population Smaller Than the Required Sample

Frequently Asked Questions About Small Eligible Populations

A Small Population Requires an Appropriate Design, Not a Convenient Calculation

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