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Margin of Error

General

Sampling Margin of Error

The range of uncertainty around a survey or sample estimate, usually expressed as '± X%', reflecting how much the result could differ from the true population value.

Definition

Margin of error is the range of uncertainty around a survey or sample estimate, most often expressed as "± X%" alongside a headline number — for example, "support stands at 52%, ± 3 percentage points." It quantifies how much the sample result might differ from the true value in the full population purely due to random sampling variation, not measurement mistakes or bias.

Margin of error is reported constantly in political polling, market research, and opinion surveys, and it is a direct input into building a Confidence Interval. A smaller margin of error means a more precise estimate, and the Margin of Error Calculator computes it directly from sample size, standard deviation (or proportion), and the desired confidence level.

Because margin of error depends heavily on sample size, researchers use the Sample Size Calculator in reverse — starting from a target margin of error and working out how many respondents are needed to achieve it.

Formula

Margin of Error = z × (σ ÷ √n)

For proportions, the formula becomes:

Margin of Error = z × √[p(1 − p) ÷ n]

Where z is the z-score for the confidence level, σ is the standard deviation, p is the sample proportion, and n is the sample size.

Worked Example

A survey of 1,000 voters finds that 52% support a proposal. Using a 95% confidence level (z = 1.96):

Margin of Error = 1.96 × √[0.52 × 0.48 ÷ 1000] = 1.96 × √0.0002496 = 1.96 × 0.01580 ≈ 0.031 (3.1%)

The reported result is therefore 52% ± 3.1%, meaning the true level of support likely falls between 48.9% and 55.1%.

Key Things to Know

  • Directly builds the Confidence Interval: the confidence interval is the point estimate plus and minus the margin of error.
  • Shrinks slowly with sample size: quadrupling the sample size is required to halve the margin of error, not merely doubling it.
  • Maximized at 50%: for proportions, margin of error is largest when the true proportion is near 50% and smaller as it approaches 0% or 100%.
  • Doesn't capture bias: a low margin of error only reflects sampling precision — it says nothing about non-response bias or Standard Deviation issues from a poorly designed survey.
  • Higher confidence means a wider margin: choosing 99% confidence instead of 90% increases the z-score and therefore widens the margin of error for the same sample.

Related Calculators

Margin of Error Calculator →Calculate the margin of error for a survey or sample estimate in seconds. Enter confidence level, standard deviation, and sample size to get the exact margin instantly.
Confidence Interval Calculator →Calculate a confidence interval from a sample mean, standard deviation, and sample size in seconds. Get the lower bound, upper bound, and margin of error instantly.
Sample Size Calculator →Find the minimum sample size needed for a survey or experiment in seconds. Enter confidence level, margin of error, and estimated standard deviation to get your answer.

Related Terms

Confidence IntervalConfidence Interval EstimateStandard DeviationStatistical Measure of Data Variability

Frequently Asked Questions

Most national political polls target a margin of error around 3% to 4% at a 95% confidence level, which typically requires sample sizes of roughly 600 to 1,100 respondents. Smaller or niche surveys sometimes accept a margin of error of 5% or more if a larger sample isn't feasible.
Margin of error is just the ± amount, such as ±3%, while the confidence interval is the full resulting range, such as 49% to 55% for a 52% estimate. The margin of error is one component used to build the confidence interval.
Margin of error is inversely proportional to the square root of the sample size, so to cut the margin of error in half you need to quadruple the sample size, not just double it. This diminishing-returns relationship is why polls rarely push sample sizes far beyond a few thousand respondents — the cost grows faster than the precision gained.
No — margin of error only captures random sampling variability and assumes a properly randomized sample. It does not account for non-response bias, poorly worded questions, or other systematic errors, which can make real-world accuracy worse than the stated margin suggests.
A 95% confidence level uses a z-score of 1.96, while 90% uses 1.645 and 99% uses 2.576. The z-score scales the standard error to produce the desired confidence level in the final margin of error and confidence interval.