Margin of Error Calculator
Enter the sample proportion, the sample size, and a z-score to get the margin of error — the ± band that says how far a poll or survey estimate might sit from the true value.
What is the margin of error?
The ± band around a survey estimate
The margin of error is the plus-or-minus band reported alongside a poll or survey result. It says how far the estimate from your sample could plausibly sit from the true value in the whole population, purely because you measured a sample rather than everyone. A poll at 47% with a margin of ±3% means the true value is plausibly between 44% and 50%. The margin depends on three things: the proportion you measured, how many people you sampled, and the confidence level you chose, expressed as a z-score.
Enter the sample proportion, the sample size, and a z-score to get the margin of error instantly.
One formula: scale the standard error of the proportion by the z-score, then turn it into a percentage.
MoE = z × √(p(1 − p) ÷ n) × 100Here p is the sample proportion written as a decimal, n is the sample size, and z is the confidence multiplier from the standard normal distribution. The term p(1 − p) is the variance of the proportion; dividing by n and taking the square root gives the standard error, and multiplying by z and 100 converts it into a percentage band. Common z-scores are 1.645 for 90% confidence, 1.96 for 95%, and 2.576 for 99%.
Suppose a poll of 1000 people finds a 50% split, and you want the 95% margin of error.
Note the proportion, sample, and confidence
The proportion is p = 0.5, the sample size is n = 1000, and 95% confidence means z = 1.96.
Find the standard error
p(1 − p) ÷ n = 0.25 ÷ 1000 = 0.00025, and √0.00025 ≈ 0.0158 — the standard error of the proportion.
Scale by z and convert to a percentage
1.96 × 0.0158 × 100 ≈ 3.1 — the margin of error is about ±3.1%, so the true value is plausibly between 46.9% and 53.1%.
The margin of error is a ± band, not a single error figure, and reading it well keeps you from over-claiming. A reported value of 47% with a ±3% margin means the true population value is plausibly anywhere from 44% to 50% at the stated confidence level — so a 47%-versus-45% gap between two options is well inside the margin and should be treated as a statistical tie, not a real lead. The size of the margin is driven by three levers. A larger sample shrinks it, but only in proportion to 1 ÷ √n, so quadrupling the sample size merely halves the margin — diminishing returns set in quickly. The proportion matters too: p = 0.5 gives the widest, most conservative margin, which is why pollsters planning a sample often assume 0.5 to guarantee a worst-case bound. Finally, a higher confidence level uses a larger z-score and therefore a wider band, so 99% confidence buys more certainty at the cost of precision. Always remember the margin captures only random sampling error; a biased sample, a leading question, or a poor response rate can push the estimate off by far more than the margin suggests.
The formula is exact, but it answers a narrow question — keep its scope in mind.
Random sampling error only, and a large-sample approximation
This margin of error measures only the random variation from sampling, not systematic bias from a non-representative sample, question wording, or non-response — those can dwarf the sampling error and are not captured here. The formula also uses the normal approximation to the binomial, which is reliable for moderate to large samples but loses accuracy for very small n or for proportions very close to 0 or 1, where exact or adjusted intervals are preferable. Treat the result as a guide to precision, not a guarantee of accuracy.