2026-08-07 10 min

Sample Size Calculator

Calculate the minimum survey sample size from confidence level, margin of error, and population proportion, with finite population correction and a reverse margin-of-error solver.

Calculator

Confidence Level

Margin of Error (%) Help

Acceptable error in points

Population Proportion (%) Help

Use 50% if not sure

Population Size (optional) Help

Enter N for FPC

Sample Size for MoE Check Help

Fixed n to reverse-check

Results update automatically

Results update live as you type

Status:95% confidence · z = 1.96n = 385MoE ±4.99%

Sample Size Summary

Required Sample Size385

FPC-Adjusted Size385

Z-Score (95%)1.96

Margin of Error at n=385±4.99%

Assuming an unlimited population.

Calculation Breakdown

Standard Error0.0255

Uncorrected MoE±4.99%

FPC-Corrected MoE—

Target MoE±5%

Results

Optimal

Required Sample Size

385people

Unadjusted n for an infinite population

FPC-Adjusted Size

385people

Assuming an unlimited population

Z-Score

1.96

Standard normal value for 95% confidence

Margin of Error (at n)

±4.99%

Achieved error with n=385

FPC-Corrected MoE

±4.99%

No finite population entered

Standard Error

0.0255

√(p(1−p)/n) for the entered sample size

Sample sizes use the Cochran normal-approximation formula with z-scores for the selected confidence level, rounded up. Finite population correction applies only when a positive population size is entered.

Visualizations

Sample Size by Margin of Error

97±10%

385±5%

1,068±3%

9,604±1%

Sample Share of Population

Sample100%

Sample as Share of Population

Coverage0%

Required vs Entered Sample

Entered n385

Formula

The Cochran formula sizes a proportion survey from the z-score of the confidence level, the estimated proportion, and the tolerable error. When the sample is a sizable fraction of a known population, the finite population correction shrinks both the required n and the achieved margin of error.

n = (z² × p × (1−p)) / e²

Alternative Form

Finite population: n' = n / (1 + (n−1)/N); Reverse: MoE = z × √(p(1−p)/n)

Variables

Symbol	Name	Unit	Description
n	Required sample size	count	Respondents to survey, always rounded up
z	Z-score	σ	Standard normal value for the confidence level
p	Population proportion	decimal	Estimated share with the trait; 0.5 is safest
e	Margin of error	decimal	Half-width of the confidence interval
N	Population size	count	Total group; blank means unlimited
n'	Corrected sample size	count	FPC-adjusted requirement for finite N
SE	Standard error	decimal	√(p(1−p)/n); precision of the estimate

Explanation

Error shrinks with the square root of n, so precision gets expensive fast: halving the error quadruples the sample. The 50% proportion maximizes p(1−p) and therefore the sample — the safe default when nothing is known.

Derivation

From the normal confidence interval half-width e = z·√(p(1−p)/n), solve for n: n = z²·p(1−p)/e². For sampling without replacement from finite N, the variance gains the factor (N−n)/(N−1); solving the corrected interval for n gives n’ = n/(1+(n−1)/N).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Look up the z-score for the chosen confidence level.

z from the standard normal table

Confidence Level:95%

Z:1.96

Intermediate: 95% confidence → z = 1.96

Result: z = 1.96

Step 2

Convert margin of error and proportion to decimals.

e = MoE/100; p = prop/100

Margin of Error:5%

Proportion:50%

Intermediate: e = 0.05, p = 0.5

Result: e = 0.05, p = 0.5

Step 3

Apply the Cochran sample-size formula for an infinite population.

n = (z² × p × (1−p)) / e²

z:1.96

p:0.5

e:0.05

Intermediate: (1.96² × 0.5 × 0.5) / 0.05²

Result: n = 385 (rounded up)

Step 4

Apply finite population correction when N is known.

n' = n / (1 + (n−1)/N)

n:385

N:unlimited

Intermediate: skipped — unlimited population

Result: n = 385

Step 5

Solve the reverse direction: margin of error for the entered sample size.

MoE = z × √(p(1−p)/n)

n:385

Intermediate: 1.96 × √(0.5×0.5/385)

Result: MoE = ±4.99%

Step 6

Shrink the margin of error with the FPC factor when applicable.

FPC = √((N−n)/(N−1))

N:—

n:385

Intermediate: no finite population

Result: Uncorrected MoE stands

Final Answer

385

Worked Examples

See how the calculator works with different scenarios.

Scenario

A store wants 95% confidence with ±5% error and no prior proportion estimate.

Inputs

Confidence:95% (z=1.96)

Margin of Error:±5%

Proportion:50%

Calculation

n = (1.96² × 0.5 × 0.5) / 0.05² = 384.16 → 385

Result

385 customers

Interpretation

The classic textbook answer: 385 completes covers nearly every general-population survey at 95/±5.

Understanding Your Result

Your Result: Required Sample Size: 385 people

Typical range: 385 for 95%/±5%, ~1,068 for ±3%, ~9,604 for ±1%. FPC matters once n exceeds ~5% of N.

What This Means

The headcount is the smallest sample that still hits your error target; the reverse solver tells you what error a fixed sample actually delivers.

Why It Matters

Under-powered surveys mislead and over-powered ones burn budget. Sizing up front is the cheapest quality control in research.

Professional Interpretation

Report the achieved margin of error from the actual n, not the planned one, and always disclose FPC use and the assumed proportion.

Recommendations

•Default to 50% proportion without pilot data
•Round sample sizes up, never down
•Apply FPC whenever N is known and small
•Budget for non-response above the computed n
•Re-check error with the final achieved n
•Prefer stratified designs for heterogeneous groups

Educational Content

Introduction

Sample size determination balances confidence, precision, and cost before fieldwork starts. This calculator implements the Cochran formula with finite population correction plus a reverse margin-of-error check.

Concept Explanation

A confidence interval’s half-width is e = z·√(p(1−p)/n). Fixing e and solving for n tells you how many respondents buy that precision; fixing n and solving for e audits a completed survey.

Applications

▸Sizing customer and employee surveys

▸Planning clinical and A/B experiment enrollment

▸Budgeting political and market-research polls

▸Auditing achieved error after fieldwork

▸Justifying methods sections in papers

▸Comparing precision across study designs

Benefits

•Prevents under-powered, misleading studies
•Quantifies the cost of tighter precision
•FPC saves interviews in small populations
•Reverse solver audits real-world samples
•Twelve confidence levels from 70% to 99.999%

Limitations

•Assumes simple random sampling
•Ignores non-response and design effects
•Proportion formula only — not means or comparisons
•No power analysis for hypothesis tests

Best Practices

•Use 50% proportion when genuinely unsure
•Inflate n for expected non-response
•Multiply by the design effect for cluster samples
•Pre-register confidence level and error target
•Report achieved, not planned, margins

Common Misconceptions

Myth: Bigger samples are always better.

Truth: Beyond the target error, extra interviews buy negligible precision while adding cost and trivial-effect significance.

Myth: 100 respondents is always enough.

Truth: Adequacy depends on error target and population — 100 gives ±9.8 points at 95%, far from ±5%.

Myth: FPC is optional polish.

Truth: Past ~5% of the population, ignoring FPC oversamples substantially.

Myth: A 95% interval has 95% chance of containing the truth.

Truth: It means 95% of such intervals from repeated sampling would capture the true parameter.

Practical Tips

•Start with 95% confidence and ±5% error
•Use 50% proportion without pilot data
•Enter N whenever the population is known
•Quadruple n to halve the error — budget accordingly
•Add 10–20% for non-response
•Re-run the reverse check with final n
•Document z, p, e, and N in your methods

Common Mistakes

Using an optimistic proportion to shrink n

Use 50% unless a solid pilot estimate exists; a wrong guess under-powers the study.

Forgetting finite population correction

Enter N for small populations — it can cut the sample by double-digit percentages.

Rounding the sample size down

Always round up: 384.16 needs 385 people, not 384.

Reporting planned instead of achieved error

Run the reverse solver on the final n and publish that margin.

Ignoring non-response

Divide the computed n by the expected response rate when inviting participants.

Assumptions

This calculator makes the following assumptions to provide estimates:

Random Sampling: Every population member has a known, equal selection chance.

Normal Approximation: The sampling distribution of the proportion is approximately normal.

Fixed Proportion: The estimated proportion holds for the whole population.

Full Response: Computed n assumes every selected subject responds.

No Design Effect: Simple random sampling; clustering or stratification needs adjustment.

Limitations

Be aware of these limitations when using this calculator:

Proportions Only: Means, totals, and comparisons need different formulas.

No Non-response Model: Bias from systematic non-response is not captured.

No Power Analysis: Hypothesis-test sizing needs effect size and power inputs.

Approximation Breakdown: Extreme proportions near 0% or 100% need exact methods.

Frequently Asked Questions

Answers to common questions about this calculator.

What is sample size?

Sample size is the number of observations or subjects included in a study, survey, or experiment — a subset of the population selected to represent the whole group. It directly affects precision, reliability, and statistical power.

Why is sample size important in research?

Too small a sample misses real effects and gives wide confidence intervals; too large wastes resources and can flag trivially small effects. Correct sizing keeps a study ethical, efficient, and valid.

What is a confidence level in statistics?

The probability that the true population parameter falls inside the calculated confidence interval — e.g. 95% means about 95 of 100 repeated surveys would capture it. Common levels are 90%, 95%, and 99%.

What is margin of error?

The range above and below the sample statistic expected to contain the true parameter at the chosen confidence level. A 60% result with ±3% at 95% confidence means the true value is plausibly 57–63%.

How does confidence level affect required sample size?

Higher confidence needs a bigger sample: at 5% margin of error and 50% proportion, 90% confidence needs n=271, 95% needs 385, and 99% needs 667, because the z-score grows (1.645 → 1.96 → 2.58).

How does margin of error affect sample size?

Halving the margin of error quadruples the sample size (inverse-square law). At 95% confidence, ±10% needs 97 respondents but ±1% needs 9,604.

What is population proportion and how do I choose it?

Your best estimate of the share of the population with the trait of interest. With no prior data, use 50% — the most conservative choice, giving the largest safe sample.

Why is 50% the most conservative choice for population proportion?

The formula contains p(1−p), which peaks at p=50% (0.25). Any other proportion shrinks the required sample, so 50% guarantees coverage whatever the truth is.

What is the sample size formula for infinite populations?

n = (z² × p × (1−p)) / e², where z is the z-score for the confidence level, p the estimated proportion as a decimal, and e the margin of error as a decimal.

What is finite population correction (FPC)?

An adjustment when the sample exceeds ~5% of the population: n’ = n / (1 + (n−1)/N). It shrinks the required sample for small populations — e.g. 385 becomes ~322 when N=2,000.

What is the relationship between sample size and margin of error?

Margin of error is inversely proportional to √n: e = z × √(p(1−p)/n). Quadrupling the sample halves the error, with diminishing returns beyond that.

What is the rule of thumb for sample size in surveys?

n=385 covers most surveys at 95% confidence with ±5% error and maximum variability. Apply finite population correction below N≈10,000.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Planning Your First Survey

From research question to sample plan.

Confidence vs Precision

Trading certainty against sample cost.

Small Populations and FPC

When the correction changes everything.

Reporting Margins Honestly

Achieved error and non-response.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Sample size: Number of observations included in a study (n).
Confidence level: Long-run capture rate of the confidence interval.
Margin of error: Half-width of the confidence interval, in points.
Z-score: Standard-normal value matching a confidence level.
Population proportion: Share of the population with the trait (p).
Finite population correction: Shrinkage applied when sampling a large fraction of N.
Standard error: Sampling variability of the estimate, √(p(1−p)/n).
Cochran formula: n = z²p(1−p)/e² for proportion surveys.
Non-response bias: Distortion when refusers differ from responders.
Design effect: Variance inflation from complex sampling designs.

Authoritative References

Trusted sources and standards referenced by this calculator.

Sample Size Determination - Wikipedia Margin of Error - Wikipedia Confidence Interval - Wikipedia Finite Population Correction - Wikipedia How to Find Sample Size - Statistics How To Sample Size Guide - SurveyMonkey Sample Size Determination - Qualtrics

Applicable Standards

ASA Ethical Guidelines for Statistical Practice

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.