Margin of Error

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Calculate

Example: a pollster interviewed 1,000 voters in a city of 100,000, and 50% said they would vote for candidate A. What is the margin of error at 95% confidence? Leave the population blank for a very large one. Change the values to work out your own case.

Sample size n Observed proportion p̂ (%) Confidence (%) Population N

Step by step

What each term means

Margin of Error (E): How far the sample proportion can be from the population proportion, at the chosen confidence level. It is the "plus or minus 3 points" of election polls: half the width of the confidence interval.
Critical Value (z_α/2): The standard normal value that leaves α/2 in each tail. At 95% confidence, z_α/2 ≈ 1.96; at 99%, ≈ 2.58. More confidence calls for a wider margin.
Standard Error of the Proportion: The standard deviation of p̂ across different samples. It peaks at p̂ = 50% and falls with the square root of n: to halve the margin, you need four times the sample.
Finite Population Correction: Shrinks the margin when the sample is a large slice of the population N. With 1,000 out of 100,000 (1%), the factor is 0.995 and changes almost nothing, which is why population size matters so little. For a very large population, it equals 1. See also the correction applied to the sample size.