Precision planning for one yes/no percentage
Sample Size for Proportion Calculator
Plan completed observations for estimating a population proportion with a two-sided margin of error. Choose an expected percentage or the conservative 50% case, then account for a finite population, design effect, and expected nonresponse.
Define the survey or study
Required field plan
At p = 0.50, the plan expects about 192.5 successes and 192.5 failures. Both exceed 10, supporting the basic normal-approximation diagnostic.
n₀ = ceil[1.9600² × 0.50 × 0.50 ÷ 0.05²] = 385
What this calculator plans
This calculator estimates the number of usable observations needed for a confidence interval around one population proportion, such as the percentage of eligible voters supporting an option, the share of shipped units with a defect, or the fraction of patients completing a follow-up. Each analyzed unit contributes a binary outcome under a clearly defined rule: yes/no, success/failure, present/absent, or another two-category indicator.
The desired margin is entered in percentage points. A five-point margin around an observed 42% proportion would produce an approximate interval from 37% to 47% under the planning method, not “5% of 42.” The formula uses decimals internally, so 5 percentage points becomes E = 0.05 and an expected 30% becomes p = 0.30.
This is a confidence-interval precision calculation, not a power calculation for comparing groups or testing a specified change. It also assumes the binary outcome definition, target population, and sampling plan are established before recruitment. A precise answer to a vague or shifting question is not useful.
The four quantities controlling n
Expected p
The variance term is p(1−p). It peaks at 0.25 when p = 0.50, making 50% the largest sample requirement for fixed confidence and margin.
Margin E
The requirement scales with 1/E². Cutting the desired half-width from five points to 2.5 points requires about four times as many simple-random observations.
Confidence
Higher confidence uses a larger two-sided z critical value. A 99% interval therefore needs more observations than a 95% interval at the same p and E.
Sampling design
A finite population can reduce the count, while clustering and unequal weighting may raise it through a design effect. Nonresponse then inflates invitations.
Formula and rounding sequence
For a large population under simple random sampling, n0 = z*²p(1−p)/E². The calculator keeps the unrounded value for downstream corrections and displays the ceiling as the initial count. If no credible estimate of p exists, select conservative 50%. If a strong prior estimate is available from the same population, mode, and outcome definition, using it can reduce n—but an optimistic extreme value can underplan the study.
For sampling without replacement from a fixed eligible population N, the finite population correction uses n = n0/[1+(n0−1)/N]. The calculator then multiplies by the design effect and rounds up to usable completes. Finally, it divides by the expected usable response fraction and rounds upward to invitations. Each stage serves a different purpose and should be reported separately.
Rounding only at the end of every mathematical step can produce small differences between software. This implementation applies corrections to the underlying unrounded large-population requirement, then takes ceilings for operational counts. It also caps invitations at N when a finite frame is entered, because one cannot invite more distinct units than exist without a replacement or repeated-contact model.
Worked example: a statewide yes/no survey
A team wants 95% confidence and a margin of ±5 percentage points. It lacks a reliable prior percentage, so it uses p = 0.50. The initial value is 1.96² × 0.50 × 0.50 / 0.05² = 384.15, which rounds up to 385 usable simple-random observations for a large population.
With design effect 1 and no finite population correction, 385 completes remain the target. If 10% of sampled units are expected not to provide a usable response, the usable response fraction is 0.90. Invitations = 385/0.90 = 427.78, so the plan calls for at least 428 invitations. The approximate margin after rounding is 1.96√[0.25/385] = 0.04994, or 4.994 percentage points.
At p = 0.50, 385 observations imply about 192.5 expected successes and 192.5 expected failures. Both are well above 10, so the familiar large-sample normal approximation is not strained by rare expected counts. This diagnostic is not a guarantee of interval quality, but it flags obvious problems when p is near zero or one and n is modest.
Expected proportion versus conservative 50%
The conservative choice protects precision across any true p because p(1−p) cannot exceed 0.25. It may recruit more units than necessary when the true percentage is near an extreme. For example, p = 0.10 produces variance 0.09, only 36% of the maximum, so the same nominal margin uses a smaller initial n.
Use an expected value only when evidence is relevant and trustworthy. A national estimate may not transfer to one state; last year’s adoption rate may not survive a policy change; an opt-in poll may not predict a probability sample. Consider scenarios at the lower and upper plausible p values. Because p and 1−p give the same variance, 20% and 80% produce identical planning n.
If the goal is to estimate several proportions, calculate each one with its required precision and use the largest valid sample. Subgroup reporting requires enough observations inside each subgroup, not merely in the overall sample. Oversampling and weights may change the design effect and therefore the total plan.
Limits of the Wald planning formula
The normal formula is transparent and widely used for planning, but the final confidence interval should not automatically be the simple Wald interval p̂ ± z√[p̂(1−p̂)/n]. That interval can perform poorly with small n or proportions near zero and one. Wilson, Agresti–Coull, or exact binomial methods may have better coverage properties depending on the setting.
Even if the final interval uses Wilson or an exact method, the conventional large-sample expression can remain a useful planning approximation. For rare events, exact simulation or software tailored to the intended interval method is preferable. The expected-success and expected-failure diagnostic displayed above helps identify when special planning is warranted.
Complex surveys require design-aware variance estimation. A single design effect is a planning summary, not a substitute for strata, clusters, weights, replicate methods, or survey software in the final analysis. If the effective sample differs across variables because of missing data, each estimate may have a different precision.
Finite populations, response, and bias
Define the frame
N should count eligible sampling units on the fixed frame. Avoid mixing people, households, records, or visits. Coverage gaps remain even when the arithmetic uses the correct N.
Response inflation protects count
Dividing by the expected usable response rate raises contact volume. It does not make respondents representative of nonrespondents or correct inaccurate answers.
Design effect needs evidence
Use prior survey estimates, anticipated cluster similarity, and weighting variation. Sensitivity calculations are more honest than an unsupported value of exactly one.
Operational loss can exceed unit nonresponse
Ineligible contacts, duplicate records, missing primary outcomes, failed quality checks, and attrition may all reduce usable n. Define which losses are included in the rate.
Reporting and related tools
Record the target proportion and coding rule, population and sampling unit, confidence level, margin in percentage points, expected p and evidence—or the conservative choice—population size, design effect, usable response rate, unrounded and rounded counts, and subgroup requirements. Freeze this plan before looking at outcomes when it supports confirmatory work.
After collection, use the verified confidence interval calculator for a general interval workflow and the binomial probability calculator for fixed-n success-count probabilities. A final one-proportion interval should use a method appropriate to the observed count and design.
Frequently asked questions
Why does 50% require the largest sample?
The Bernoulli variance p(1−p) reaches its maximum 0.25 at p = 0.50. Moving toward zero or one lowers that term and the normal-formula sample requirement.
Is a 5% margin the same as five percentage points?
Not always in ordinary speech. This calculator explicitly uses percentage points. Around 40%, five points means an interval half-width of 0.05, not 5% of 40%.
Can I plan a two-group comparison here?
No. Comparing independent or paired proportions requires allocation, a target difference, alpha, power, and assumptions for both groups. This calculator estimates one population proportion.
Should I round up?
Yes. Every operational requirement is rounded upward. Rounding down can make the nominal precision slightly worse than requested.
What if expected successes are below 10?
Treat the normal approximation as questionable. Consider exact-binomial or interval-specific planning, revise the expected p, or increase n. The threshold is a diagnostic convention, not a theorem.
Does a huge sample remove survey bias?
No. It can make a biased estimate very precise. Frame coverage, selection, nonresponse, question wording, measurement, processing, and weighting still determine credibility.
References
The core planning expression follows the U.S. National Institute of Standards and Technology guidance on selecting sample sizes, which gives n = z²pq/δ² under a normal approximation. NIST also discusses confidence intervals for proportions, including exact and Wilson-related approaches that may be preferable for final inference. Save the unrounded calculation and every planning assumption so independent reviewers can reproduce the field target and distinguish precision choices from later operational losses.