ConfidLab

Margin of Error Calculator

How much a poll result could be off, purely from sampling.

Sample proportion (0–1)

Sample size (n)

Confidence level (%)

Margin of error

± 4.9%

Reported result: 50% ± 4.9%

What you entered

Proportion: 50%n: 400Confidence: 95%

1
Sample proportion variance: p(1−p)
0.5 × (1 − 0.5)= 0.25
2
Standard error
√(0.25 ÷ 400)= 0.025
3
Margin of error
1.96 × 0.025= 0.049

Result

Margin of error: ± 4.9%

With a sample of 400 at 50%, the margin of error is ± 4.9 percentage points at 95% confidence.

Margin of error at 95% confidence, worst-case proportion

These use p = 0.5, which maximises the margin and is what pollsters quote when the true proportion is unknown.

Sample size	Margin of error	As a percentage
100	0.0980	9.80%
250	0.0620	6.20%
384	0.0500	5.00%
500	0.0438	4.38%
1,000	0.0310	3.10%
2,000	0.0219	2.19%

The 384 row is why so many published polls use a sample of about 1,000 or quote plus or minus 3%: 384 is the smallest sample giving a 5% margin at 95% confidence, and 1,000 gets you to about 3%. Halving the margin requires quadrupling the sample, so the step from 3% to 1.5% needs 4,000 respondents. Note what this does not cover: it is sampling error only, and says nothing about bias from who answers, how questions are worded, or who was reachable in the first place - which in practice are often larger sources of error than the margin quoted.

Why every poll reports a ± number

A poll of 1,000 people can't perfectly represent millions of voters — the margin of error quantifies how much the reported percentage could plausibly differ from the true population value, just from random sampling variation.

The 50% worst case

Margin of error is largest when the true proportion is near 50% and shrinks as it moves toward 0% or 100% — this is why pollsters often report worst-case margins assuming a 50/50 split.

Frequently asked questions

A poll of 1,000 voters shows 52% support for Candidate A. Is that a real lead?

At 95% confidence, MoE = 1.96 × √(0.52×0.48/1000) = ±3.1%. The true support is likely between 48.9% and 55.1%. Since the range includes values below 50%, the lead is NOT statistically certain — the race is too close to call.

Why does a poll of 1,000 people represent millions?

Random sampling works because of the law of large numbers: a properly random sample of ~1,000 captures the population's views within ±3% at 95% confidence, regardless of whether the population is 100,000 or 300 million. Sample size matters; population size barely does.

How do I reduce the margin of error?

Three ways: (1) Increase sample size — quadruple n to halve MoE. (2) Lower confidence level — from 99% to 95% narrows the interval by ~24%. (3) Target a proportion far from 50% — a 90/10 split has a much smaller MoE than 50/50 at the same n.