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Calculator Answers

Margin of Error Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A margin of error calculator is a statistical utility that computes the maximum expected difference between a sample result and the true population value. It uses the sample size, the standard deviation or sample proportion, and the desired confidence level (typically ninety-five percent) to calculate the margin. The resulting value expresses the precision of a survey or poll, indicating the range within which the true population parameters likely fall. Researchers, pollsters, and students use this tool to evaluate data reliability.

Pick a confidence level, enter the sample size and observed proportion, and the calculator returns the margin of error plus the standard error and z-score used. Add a population size for the finite-population correction.

See also: Confidence Interval Calculator, P Value Calculator, and Standard Deviation Calculator – Variance & SD.

Quick Answer

Calculate the margin of error for your survey or poll. Enter your sample size, population size, and confidence level to determine statistical precision.

Confidence level

Number of respondents in the sample. · e.g. 1000

%

Use 50% when the true proportion is unknown. · e.g. 50

Apply a finite-population correction. Leave blank for large or unknown populations. · e.g. 50000

Margin of error shrinks with bigger samples and grows with higher confidence. Defaulting the proportion to 50% gives the most conservative (largest) margin of error.

Margin of error

95% confidence on 1,000 respondents

±3.1%

Range: 46.9% to 53.1%

Z-score used1.96
Standard error0.0158
Margin of error (decimal)0.031
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Examples

1,000 respondents, 95% confidence, p = 50%

≈ ±3.10% (range 46.90% to 53.10%)

400 respondents, 95% confidence, p = 60%

≈ ±4.80%

1,000 respondents, 99% confidence, p = 50%

≈ ±4.07%

How it works

The standard error of a sample proportion is the square root of p(1 − p) ÷ n. Multiplying by the z-score for the chosen confidence level gives the margin of error. Adding a population size brings in a finite-population correction that shrinks the margin when the sample is a meaningful fraction of the population.

Standard error · SE = √(p(1 − p) ÷ n)

Margin of error · MoE = z × SE

FPC · √((N − n) ÷ (N − 1))

Z-scores: 90% = 1.645 · 95% = 1.96 · 99% = 2.576

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Confidence Interval CalculatorCalculate a confidence interval for a mean or a proportion. Returns the lower and upper bounds, margin of error, and standard error.P Value CalculatorConvert a z-score to a p-value for a one-tailed or two-tailed test, with the area on each side of the curve.Standard Deviation Calculator – Variance & SDUse this sd calculator to compute sample and population standard deviation, variance, mean, and count from a list of numbers.

Frequently asked questions

Margin of error is a confidence-bound around a sample estimate. A poll with a margin of error of plus or minus 3 percent at 95 percent confidence says, in plain terms, that if the poll were repeated many times under the same conditions, about 95 percent of those polls would land within 3 percentage points of the true population value. It does not mean every poll is within that range, and it ignores non-sampling errors like wording bias or response refusal.

First compute the standard error: the square root of p × (1 − p) divided by sample size, where p is the observed proportion. Then multiply by the z-score for your confidence level (1.645 for 90 percent, 1.96 for 95 percent, 2.576 for 99 percent). The product is the margin of error in decimal form; multiply by 100 to express as a percent.

Sample variance p × (1 − p) is largest when p = 0.5, which makes the margin of error largest at that point. Using 50 percent gives the most conservative (worst-case) margin of error, which is the standard convention when the true proportion is unknown ahead of polling.

When sample size is a meaningful fraction of the total population, the standard formula overstates the margin of error. The finite-population correction (FPC) multiplies the margin by the square root of (N − n) divided by (N − 1), where N is the total population and n is the sample size. It only changes the result meaningfully when the sample is more than about 5 percent of the population.

Margin of error shrinks as sample size grows, but the relationship is not linear. Doubling the sample size shrinks the margin by a factor of roughly the square root of 2, or about 30 percent. Going from 400 to 1,600 respondents cuts the margin roughly in half. Once samples are large (say, several thousand), the marginal value of adding more drops off.

No. This calculator is for planning, education, and quick sanity checks. Real survey design has to consider question wording, sampling frame, response rate, weighting, design effect from clustered or stratified samples, and non-sampling error. For formal research use professional statistical software or consult a qualified statistician.