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Sample Size Calculator | Find required sample size for confidence & margin

Determine how large a survey sample you need for a chosen confidence level, margin of error and population size. Uses the standard sample-size formula with the 50% worst-case assumption explained — essential for planning polls, experiments and market research.

Last updated: June 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

Sample size for a large population (50% response):

Margin of error90% conf.95% conf.99% conf.
+/-5%271385664
+/-3%7521,0681,843
+/-2%1,6912,4014,147
+/-1%6,7649,60416,588

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For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

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Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

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Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

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Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

What changes the required size:

LeverEffect
Lower margin of errorNeeds a larger sample
Higher confidenceNeeds a larger sample
Response near 50%Needs the largest sample (most variance)
Small populationApply the finite population correction

lightbulb Worked example

Let's say you are using the Sample Size Calculator. Determines how many survey respondents you need for statistically reliable results. Enter your population size, desired confidence level — 90%, 95%, or 99% — and the margin of error you can accept, Enter the values that match your situation into the input fields and press calculate — using realistic numbers makes the result directly useful for you.

Result: The calculator instantly applies the formula n = z^2 x p(1-p) / e^2; then correct for finite population: n_adj = n / (1 + (n-1)/N) and returns the result with appropriate precision.

What this means: Read the result in the context of what you are measuring. The step-by-step breakdown lets you confirm the math and understand which input most affects the outcome.

Formula & methodology

Formula: n = z^2 x p(1-p) / e^2; then correct for finite population: n_adj = n / (1 + (n-1)/N)

The Sample Size Calculator is built on a well-established calculation method. It uses the formula n = z^2 x p(1-p) / e^2; then correct for finite population: n_adj = n / (1 + (n-1)/N) to turn your inputs into a reliable result. Determines how many survey respondents you need for statistically reliable results. Enter your population size, desired confidence level — 90%, 95%, or 99% — and the margin of error you can accept, and the tool returns The steps are shown on the page so you can follow the reasoning from input to output.

The calculator finds the minimum number of responses needed for results within your chosen margin of error at a given confidence level. It uses the standard formula with p = 0.5 (the most conservative, highest-variance assumption) unless you supply an expected proportion. For small or known populations it applies the finite population correction so you don't over-sample. Tighter margins and higher confidence both demand larger samples.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How big should my survey sample be?
The Sample Size Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Determines how many survey respondents you need for statistically reliable results. Enter your population size, desired confidence level — 90%, 95%, or 99% — and the margin of error you can accept, and the tool returns the minimum sample size required, plus There is no limit on usage, and it works on any device.
What is margin of error?
The margin of error is the maximum expected difference between your sample statistic and the true population value, at a chosen confidence level — usually ±5% at 95% confidence. It comes from the standard error of the estimate, scaled by the critical z-value. A 5% margin means the true value is expected to lie within 5 points of your result 95 times out of 100.
Why assume a 50% response rate?
The 50% assumption is the worst case: p = 0.5 maximizes the variance p(1−p) in the sample-size formula, producing the largest required sample. It is used when the true proportion is unknown, because any actual value below 50% needs a smaller sample. This conservative choice guarantees your sample is large enough regardless of the real response distribution.
Does population size matter?
Surprisingly little for large populations: the sample size needed to achieve a given margin of error barely changes whether the population is 10,000 or 10 million, because the standard error depends mainly on the sample itself. Population size only starts to matter for small populations (under a few thousand), where the finite population correction reduces the required sample.
How does confidence level affect sample size?
Enter the required values into the input fields and press the calculate button. The tool applies the formula n = z^2 x p(1-p) / e^2; then correct for finite population: n_adj = n / (1 + (n-1)/N) and returns the result with a short explanation of each step, so you can check the working yourself.
What does the Sample Size Calculator do?
Determines how many survey respondents you need for statistically reliable results. Enter your population size, desired confidence level — 90%, 95%, or 99% — and the margin of error you can accept, and the tool returns the minimum sample size required, plus. The calculator takes your inputs, applies the standard calculation, and returns a clear result so you can make an informed decision without doing the math by hand.
What formula does the Sample Size Calculator use?
The Sample Size Calculator applies the standard formula: n = z^2 x p(1-p) / e^2; then correct for finite population: n_adj = n / (1 + (n-1)/N). The tool walks through each step of the calculation so you can verify the numbers yourself.
Is the Sample Size Calculator free and private?
Yes — the Sample Size Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
How accurate is the Sample Size Calculator?
The Sample Size Calculator applies the standard formula: n = z^2 x p(1-p) / e^2; then correct for finite population: n_adj = n / (1 + (n-1)/N). It is tested against published worked examples before launch and uses native double-precision arithmetic, so results are accurate to typical precision for the values you enter — with no intermediate rounding that could distort the answer.

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