Margin of Error Calculator

Calculate margin of error for surveys and polls including confidence intervals and sample size analysis.

MOE = Z * sqrt(p(1-p)/n). For finite populations: MOE = Z * sqrt(p(1-p)/n) * sqrt((N-n)/(N-1)). Where Z is the Z-score, p is proportion, n is sample size, N is population size.
For a sample of 1,000 people at 95% confidence (Z=1.96) with p=0.5: MOE = 1.96 * sqrt(0.5*0.5/1000) = +/-3.1%. Confidence interval: 46.9% to 53.1%.

What is margin of error?

Margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It represents the range in which the true population value is likely to fall. For example, a margin of error of +/-3% at 95% confidence means we can be 95% confident that the true population value falls within 3 percentage points of the survey result.

How do you calculate margin of error?

The margin of error formula is: MOE = Z * sqrt(p(1-p)/n), where Z is the z-score for your confidence level (1.96 for 95% confidence), p is the sample proportion, and n is the sample size. For finite populations, multiply by the finite population correction: sqrt((N-n)/(N-1)), where N is the population size.

What is a good margin of error?

A good margin of error depends on your needs, but for most surveys, +/-3-5% at 95% confidence is considered acceptable. Political polls often aim for +/-3%, while academic research might require +/-5% or less. The acceptable margin depends on the required precision, available resources, and consequences of inaccuracy.

What are confidence levels and Z-scores?

Confidence level indicates how certain you are that the true value falls within the margin of error. Common confidence levels and their Z-scores are: 90% (Z=1.645), 95% (Z=1.96), and 99% (Z=2.576). A 95% confidence level means if you repeated the survey 100 times, about 95 would produce results within the margin of error.

How does sample size affect margin of error?

Sample size has an inverse square root relationship with margin of error. To cut the margin of error in half, you need to quadruple the sample size. Larger samples give more precise estimates (smaller margins of error), but gains diminish as sample size increases. For example, increasing from 100 to 400 people cuts the margin in half, but going from 1,000 to 4,000 also only cuts it in half.

Why use 0.5 for proportion when calculating margin of error?

Using a proportion of 0.5 (50%) gives the maximum possible margin of error for a given sample size and confidence level. This is conservative planning - if you don't know what the result will be, using 0.5 ensures you won't underestimate the margin of error. The margin is smallest when the proportion is near 0 or 1, and largest at 0.5.