Sample Size Calculator

What this calculator does

This calculator estimates how many responses a survey needs to reach a desired margin of error at a chosen confidence level. It is designed for proportion questions, such as the share of customers who prefer a feature or the share of voters who support a candidate.

The result is the minimum sample size, rounded up to a whole person or response. Smaller margins and higher confidence levels require larger samples. A finite population correction can reduce the requirement when the population is small enough that sampling without replacement noticeably changes the variance.

How to use this calculator

1. Choose a confidence level from 70% to 99.999%.
2. Enter the margin of error as a percentage, such as 5 for plus or minus five percentage points.
3. Enter the expected population proportion as a percentage. Use 50% if you have no prior estimate, because that value produces the most conservative sample size.
4. Optionally enter the population size. Leave it blank for an effectively unlimited population.
5. Click Calculate.

The headline shows the required sample size. The rows below show the z-score, the unadjusted estimate, and whether the finite population correction was applied.

Formulas

Unadjusted sample size: n₀ = z² × p × (1 − p) / e²

Finite population correction: n = n₀ / (1 + (n₀ − 1) / N)

Here z is the standard normal value for the chosen confidence level, p is the population proportion as a decimal, e is the margin of error as a decimal, and N is the population size. The final result is the ceiling of n, because a fraction of a response cannot be collected.

Default example

The default inputs are a 95% confidence level, a 5% margin of error, and a 50% population proportion with no population limit. The z-value is about 1.96, so the unadjusted estimate is 1.96² × 0.5 × 0.5 / 0.05², which equals about 384.15.

Rounding up gives a required sample size of 385. If the population is limited to 1,000 people, the finite correction lowers the requirement to 278 responses. The smaller population means each response carries more information, so fewer are needed for the same precision.

Assumptions and limitations

• The formula assumes a simple random sample. Complex designs such as stratified, cluster, or quota sampling usually need a larger sample or a design-effect adjustment.
• The normal approximation to the binomial distribution is assumed. This works well for typical margins and proportions but can be unreliable for very small samples or proportions near 0% or 100%.
• The population proportion must be strictly between 0% and 100%. A proportion of exactly 0% or 100% would imply no variability and is outside the model.
• The finite population correction assumes sampling without replacement from a known, fixed population.
• The calculator estimates the sample size needed for one proportion. It does not account for nonresponse, invalid responses, or subgroup analyses, all of which may require a larger fielded sample.
• The margin of error is a percentage-point radius, not a percentage of the measured value.

Frequently asked questions

Why does a 50% proportion give the largest sample size?

The quantity p × (1 − p) is largest when p is 0.5. A proportion near 0% or 100% makes the outcome more predictable, which reduces the variability and therefore the sample size needed. Using 50% is the safe choice when the true proportion is unknown.

What does the margin of error mean?

The margin of error is the radius of the confidence interval around the observed proportion. A 5% margin at 95% confidence means that if the survey were repeated many times, about 95% of the resulting intervals would contain the true population proportion.

Do I still need 385 responses if I know the population size?

Not necessarily. When the population is finite, the correction formula can reduce the requirement, as the 1,000-person example shows. If the population is very large relative to the sample, the correction has little effect and the unlimited-population estimate is close enough.

Reference table

Z-scores by confidence level

Use the z-score that matches the confidence level. Higher confidence requires a larger sample because the margin of error must be smaller.

Source

Reference: Calculator.net — Sample Size Calculator.