# Sample Size Calculator

> Find how many responses a survey needs to estimate a proportion within a chosen margin of error and confidence level, with a finite population correction.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/sample-size-calculator

Choose how confident you want to be and how close the estimate must be, and get the number of completed responses needed for a survey percentage.

Also called: Survey sample size calculator.

## Inputs
- Confidence level (unit %; default 95)
- Margin of error (±) (unit %; default 5)
- Expected proportion (unit %; default 50)
- Population size (optional) (default ; optional)

## Results
- Responses needed
- Needed for an unlimited population
- z-score for the confidence level
- Exact value before rounding up

## Formula

```
n₀ = z² × p(1 − p) ÷ E²;  with population N: n = n₀ ÷ (1 + (n₀ − 1) ÷ N);  round up
```

Where:
- z: standard normal value for the two-sided confidence level (1.96 for 95%)
- p: expected proportion
- E: margin of error as a decimal
- N: population size

## Worked example
1. For 95% confidence, z = 1.959964.
2. n₀ = 1.959964² × 0.5 × 0.5 ÷ 0.05² = 3.841459 × 0.25 ÷ 0.0025 ≈ 384.15.
3. Round up: 385 responses.

**Result:** To estimate a percentage within ±5 points with 95% confidence, you need 385 completed responses.

## What the result means

This is the number of completed responses, not invitations. If you expect a 20% response rate, invite about five times as many people.

## Assumptions and limits
- A simple random sample, estimating a single proportion, using the normal approximation to the binomial distribution.
- Clustered or stratified designs need a design effect; comparing two groups or estimating a mean needs a different formula.
- A random sample only fixes random error, not bias from who chooses to respond.

## Questions

### Why does a population of millions need only about 385 people?

Precision depends mostly on the sample size, not the population. The population matters only when the sample is a sizeable share of it, which the finite population correction handles.

### Why use 50% for the expected proportion?

p(1 − p) is largest at 50%, so it gives the most conservative (largest) sample. If you know the answer is near 10% or 90%, a smaller sample suffices.

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Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/sample-size-calculator