Margin of Error Calculator

Find the margin of error and confidence interval for a poll result or an average, with an optional correction for small populations.

Your data

Confidence level (%)

Sample size (n)

Sample proportion (p̂)

Population size (optional)

Leave the population size blank unless the sample is a large share of a small population. Using p̂ = 0.5 gives the largest possible margin of error.

Critical value (z)

1.96

Standard error

0.01581

Margin of error

±3.1%

95% confidence interval: 46.9% to 53.1%

SE = √[p̂(1 − p̂) ÷ n] = √[0.5 × 0.5 ÷ 1000] = 0.01581

Margin of error = z × SE = 1.96 × 0.01581 = 0.03099 (3.1 percentage points)

How to use this tool

Choose proportion (a percentage from a poll or survey) or mean (an average when the population standard deviation is known).
Enter the confidence level, usually 95, and the sample size.
Enter the sample proportion, or the standard deviation and sample mean.
Optionally enter the population size if your sample is a big share of a small population.

Formula Reference

Margin of error

ME = z × SE

SE of a proportion

SE = √[ p̂(1 − p̂) ÷ n ]

SE of a mean (σ known)

SE = σ ÷ √n

Finite population correction

√[ (N − n) ÷ (N − 1) ]

CalcConfidence Interval Calculator CalcSample Size Calculator ArticleConfidence Intervals Explained ArticleThe Finite Population Correction ArticleStandard Error GlossaryGlossary: Margin of Error

Frequently Asked Questions

What does a margin of error of ±3% mean?+

It means that, in repeated polls of the same size, the sample percentage would land within about 3 percentage points of the true population percentage in the stated share of polls, usually 95%. It describes sampling variability only, not bias from a poor sample or badly worded questions.

Why is p̂ = 0.5 the worst case?+

The product p̂(1 − p̂) is largest when p̂ is 0.5, so that value gives the widest margin of error. Pollsters plan sample sizes with 0.5 to be safe.

Does this work when the population standard deviation is unknown?+

Not exactly. This calculator uses the normal (z) critical value, which is right for proportions and for means with known σ. For a mean with unknown σ and a small sample, use the t-based Confidence Interval Calculator, which gives a slightly wider interval.

How can I make the margin of error smaller?+

Increase the sample size, lower the confidence level, or sample a larger share of a small population. Because the margin shrinks with the square root of n, quadrupling the sample size halves it.

How to use this tool

Choose proportion (a percentage from a poll or survey) or mean (an average when the population standard deviation is known).

Enter the confidence level, usually 95, and the sample size.

Enter the sample proportion, or the standard deviation and sample mean.

Optionally enter the population size if your sample is a big share of a small population.