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Sample Size Calculator

Determine the minimum sample size for statistically valid surveys and experiments — with live power analysis and scenario comparison.

Parameters

Cohen's d / h conventions: Tiny=0.1, Small=0.2, Medium=0.5, Large=0.8
Typical: 3% (precise) · 5% (standard) · 10% (rough)
Use 0.5 for worst-case (largest n)
Enables finite population correction (FPC) when population is known
Required Sample Size
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Enter parameters to calculate
Required n
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Confidence Level
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Margin of Error
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Statistical Power
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Degrees of Freedom
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Population
Infinite
n = z²×p(1−p)/E² n_adj = n×N/(n+N−1)
Conservative — 99% CI
High confidence · Most samples needed
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subjects required
Standard — 95% CI
Industry standard · Recommended
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subjects required
Base scenario
Relaxed — 90% CI
Lower confidence · Fewest samples
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subjects required

Sample Size Matrix — Confidence × Margin of Error

Required n (proportion study, p = 0.5, infinite population). Gold outline = your current settings.

MoE 90% CI 95% CI 99% CI

Population Size Impact

How finite population correction reduces required n at current parameters (95% CI).

Population Size Required n % of Pop. Reduction vs. Infinite

Sample Size vs. Margin of Error

How required n changes across margins of error for all three confidence levels (p = 0.5).

Chart: sample size vs. margin of error.

Quick-fill common research scenarios:

Achievable Power

Given a fixed sample size, what statistical power can you achieve?

Uses the Confidence Level and Effect Size from the Calculator tab.

Minimum Detectable Effect

What’s the smallest effect detectable with a given sample at 80% power?

Computes at 80% target power with current Confidence Level.

Power vs. Sample Size by Effect Size

How statistical power grows as n increases, for small, medium, and large effects. Dashed line = 80% power target.

Chart: power vs. sample size by effect size.

Rule of Thumb for Common Research Scenarios

Research ScenarioTypical nCINotes
Pilot study30–5090%Feasibility testing only
Exploratory survey100–20090–95%Rough estimates, ±7–10%
Standard opinion poll400–60095%±5% margin of error
National-scale survey1,000–1,50095%±3% margin of error
Clinical trial (Phase II)50–30095%Power usually 80–90%
A/B test (5% lift)~1,500/group95%Depends on baseline rate
Regression analysis10–20× predictors95%Minimum 10:1 rule
Usability testing5–8 per user typeN/AQualitative, not statistical
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How to Use This Calculator

1

Choose Your Study Type

Select Proportion Study for surveys measuring percentages (e.g. approval rates). Select Mean Study when estimating an average value (e.g. blood pressure, test scores).

2

Set Confidence & Margin of Error

95% confidence and ±5% margin of error are industry-standard defaults. Increase precision (lower MoE) or increase confidence (99%) to see how sample requirements change.

3

Review Power Analysis

Select an effect size chip to see the statistical power your sample achieves. Aim for ≥80% power. The power curve chart shows how adding subjects increases your ability to detect real effects.

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Formula & Methodology

Sample Size for Proportions
n = z² × p(1−p) / E²
Where z is the critical value for your confidence level (1.645 / 1.96 / 2.576 for 90/95/99%), p is the expected proportion, and E is the margin of error. Use p = 0.5 for the most conservative (largest) estimate.
Sample Size for Means
n = (z × σ / E)²
Where σ is the estimated population standard deviation and E is the desired margin of error in the same units. Requires a pilot estimate or prior study for σ.
Finite Population Correction
n_adj = n × N / (n + N − 1)
When the sample represents more than 5% of the population, this correction reduces the required sample size. Significant for small populations (under 10,000).
Statistical Power
Power = Φ(d√n − zα)
Where d is Cohen's effect size, n is sample size, zα is the critical value, and Φ is the standard normal CDF. A well-powered study has Power ≥ 0.80.

About this calculation · Accuracy & Methodology

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Key Terms

Sample Size (n) The number of observations or responses required to achieve the desired statistical precision.
Margin of Error (E) The maximum expected difference between the sample statistic and the true population parameter. Halving the margin of error roughly quadruples the required sample size.
Confidence Level The percentage of repeated intervals that would contain the true parameter. A 95% CI means 95 out of 100 such intervals capture the truth.
Statistical Power (1−β) The probability of detecting a real effect when one exists. Conventional minimum is 80%. Low power risks Type II errors (false negatives).
Effect Size (Cohen's d) A standardised measure of the magnitude of a difference or relationship. Larger effects are easier to detect and require smaller samples. Small = 0.2, Medium = 0.5, Large = 0.8.
Finite Population Correction A mathematical adjustment that reduces required sample size when the population is small relative to the sample. Applied when n/N > 5%.
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Real-World Examples

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National Political Survey

Input: 95% CI, E = ±3%, p = 0.5

Result
n = 1,068

n = 1,068 — minimum for a credible national poll. Note: population of 330M barely affects this number.

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Designing Studies with the Right Sample Size

Why Sample Size Matters

An undersized study lacks the statistical power to detect real effects — wasting resources and producing false negatives. An oversized study is unnecessarily expensive and, in clinical research, may expose more participants to experimental conditions than needed. Proper sample-size calculation balances precision, power, and practical constraints before data collection begins.

The Diminishing Returns of Larger Samples

The margin of error shrinks with the square root of the sample size. Quadrupling n only halves the margin. Going from 400 to 1,600 respondents halves precision — but the cost quadruples. Most researchers find a practical sweet spot where precision is sufficient without being prohibitively expensive.

Power Analysis: The Often-Forgotten Step

Many researchers focus solely on confidence and margin of error, ignoring power. A study can have a tight 95% CI and still be underpowered — meaning it's well-suited to describe what it observed, but not sensitive enough to detect small effects. Power analysis (the Power Analysis tab) lets you work backwards: given your budget (fixed n), what is the smallest effect you can reliably detect?

Finite Populations and the Census Myth

A common misconception is that you need to survey a large fraction of a population to get reliable results. In reality, for populations above ~10,000, the required sample size barely changes. The precision of a survey is determined almost entirely by the absolute size of the sample, not by what fraction of the population it represents.

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Frequently Asked Questions

General What sample size do I need for 95% confidence and 5% margin of error?
For a proportion with p = 0.5 (worst case): n = (1.96² × 0.5 × 0.5) / 0.05² = 384. This is the classic "standard survey" benchmark. For a finite population of 10,000, the FPC reduces it to about 370.
General Does population size always affect the sample size I need?
Only when your sample is large relative to the population (n/N > 5%). For populations above 100,000, the standard infinite-population formula works fine — the difference is under 1%. This is why a poll of 330M people uses only ~1,000 respondents.
Tips What proportion should I use if I don't know the true value?
Use p = 0.50. This maximises variance (p × (1−p) is highest at 0.5) and gives the most conservative — therefore safest — estimate. Any other value assumes knowledge you don't have and could leave your study underpowered.
Basics What is statistical power and why should I care?
Power (1 − β) is the probability of detecting a real effect. At 80% power, you have a 20% chance of missing a real difference (Type II error). Studies with power below 60% are often not worth conducting — they're likely to produce inconclusive results even when a real effect exists.
Basics What is Cohen's effect size?
Cohen's d measures the standardised magnitude of a difference: d = (μ₁ − μ₂) / σ. Conventions: Small = 0.2 (like height difference between 15/16-year-olds), Medium = 0.5 (like IQ difference between college and high-school groups), Large = 0.8 (obvious differences). For proportions, Cohen's h is analogous.
Method Can I use the same formula for means and proportions?
No — they use different formulas. Proportions use n = z²p(1−p)/E². Means use n = (zσ/E)². For means, you must supply an estimate of the population standard deviation (σ), typically from a pilot study or published literature.
Advanced Why does 99% CI require so many more samples than 95%?
Because the z-critical value scales non-linearly: z₉₅ = 1.96 but z₉₉ = 2.576. Since n scales with z², the 99% CI needs (2.576/1.96)² ≈ 72% more samples than 95% for the same margin of error. The jump from 95% to 99% is much larger than from 90% to 95%.
Basics What is the minimum sample size for valid statistics?
A common rule of thumb is n ≥ 30, which is roughly where the Central Limit Theorem kicks in and the normal approximation becomes reliable. For proportions, you also want np ≥ 10 and n(1−p) ≥ 10. For very small samples, use t-distribution methods or exact tests.
Method How do I reduce my required sample size?
Four levers: (1) Widen margin of error — doubling E reduces n by 4×. (2) Lower confidence level from 99% to 95% or 90%. (3) Use a known proportion closer to 0 or 1 instead of 0.5 (if justified). (4) Apply finite population correction if your population is small. Each lever has trade-offs in precision or confidence.
Basics What is Type I vs. Type II error?
A Type I error (α) is a false positive — concluding an effect exists when it doesn't. The confidence level controls this: 95% CI means α = 5%. A Type II error (β) is a false negative — missing a real effect. Power = 1 − β. These are inversely related: increasing one typically increases the other for a fixed sample size.
Method How do I choose the right margin of error?
Consider the decision at stake: ±10% is sufficient for rough feasibility checks; ±5% is standard for most business surveys; ±3% is typical for political polls; ±1% is needed for high-stakes regulatory studies. Always balance precision against cost — collecting 4× more data to halve the margin of error is rarely justified unless decisions are very sensitive to that precision.
Advanced Why do national polls use such small samples?
Because required sample size depends mostly on desired precision, not population size. For a population of 330 million vs. 10,000, the difference in required n (at 95%, ±3%) is only ~0.03%. This counterintuitive result comes directly from the finite population correction formula: when N is huge, n/(n+N−1) ≈ 0 and correction is negligible.