Margin of Error Calculator

Table of contents

• How to use
• Common mistakes
• Definitions
• Methodology
• Limitations & assumptions
• Sources

Margin of error: The plus-or-minus amount that estimates sampling uncertainty around a survey percentage at a selected confidence level.

Percentage point: The direct difference between percentages. Moving from 42% to 45% is an increase of 3 percentage points.

Confidence level: The long-run level used to build the interval. A higher confidence level gives a wider margin when other inputs stay the same.

Confidence interval: The range from the lower confidence bound to the upper confidence bound around the entered survey result.

Design effect: A multiplier that adjusts estimated sampling spread for survey features such as weighting or clustering. A design effect of 1 uses the simple random sample assumption [2].

Finite population correction: An adjustment used when a sample comes from a known, closed group and includes a sizable share of that group.

Effective sample size: The completed-response count divided by the design effect. It is a diagnostic count with similar calculated precision under a simple random sample assumption.

Mistake: Entering invitations sent, clicks, or opened emails as "Completed responses."
Fix: Enter only completed responses used to calculate the reported percentage.

Mistake: Using the full survey count for a subgroup percentage.
Fix: Use the completed-response count for that subgroup, because its sampling margin depends on its own count.

Mistake: Reading a 3-point margin as a 3% relative change.
Fix: Treat it as 3 percentage points. A 42% result with a plus-or-minus 3-point margin has bounds near 39% and 45%.

Mistake: Keeping 95% confidence when the report uses 90% or 99%.
Fix: Match the confidence level in the survey methodology or assignment.

Mistake: Entering an undocumented design effect.
Fix: Leave it blank or enter 1 unless the survey methodology, weighting report, or statistician provides a value.

• This calculator estimates sampling variability for a survey percentage or an average value, or plans sample size under the same normal approximation. It does not measure nonresponse, coverage, selection, question wording, processing, or other systematic errors.
• It uses a normal approximation. At 0% or 100%, the plug-in formula gives a zero margin, which can understate uncertainty for rare outcomes or small samples.
• Use a design effect only when it is documented for the survey. A value above 1 widens the margin and lowers the effective sample size.
• Enter population size only for a known, closed population. The population must be at least the completed-response count.
• Confidence bounds are not clipped to 0% or 100%. With a small sample or an extreme percentage, the normal approximation can produce bounds outside that range.

Calculation method

The calculator changes the entered survey percentage into a proportion, estimates its sampling spread, and multiplies that spread by a normal critical value. The two-sided critical values used are about 1.645 for 90%, 1.960 for 95%, and 2.576 for 99% confidence.

p = survey result / 100

SE = 100 x sqrt(DEFF x p x (1 - p) / n) x sqrt((N - n) / (N - 1))

In this formula, p is the survey result as a decimal, n is completed responses, DEFF is the design effect, and N is the known population size. The final square-root factor is used only when a population size is entered; otherwise it equals 1. SE is measured in percentage points.

margin of error = critical value x SE

lower bound = survey result - margin of error; upper bound = survey result + margin of error

effective sample size = n / DEFF

Average values and sample-size planning

For "Average value", standard error = standard deviation / sqrt(n), with the same optional population correction. Margin = z × standard error. Use a known population standard deviation for a mean interval; an estimated value supports approximate planning but does not account for the extra uncertainty required by a small-sample t interval. See NIST confidence-interval guidance.

For "Sample size needed", first solve the same formula for n. For percentages, n0 = z² × DEFF × p × (1 − p) / E², where E is the target margin divided by 100. For averages, n0 = (z × standard deviation / E)², with E in value units. For a known population N, n = N × n0 / (N − 1 + n0). The selected rounding is then applied and the achieved margin recalculated. These counts mean completed observations, not invitations. The optional "Design effect (multiplier)" applies to the survey-percentage calculation, not the average-value calculation.

For example, 95% confidence, an expected 50% result and a target margin of 5 percentage points require 385 completed responses with no population correction and design effect 1. For an average with known population standard deviation 12 and 100 measurements, the 95% margin is about 2.35 value units.

Worked example

With 1,000 completed responses, a 50% survey result, 95% confidence, no population correction, and design effect 1, the standard error is about 1.58 percentage points. Multiplying by 1.960 gives a margin of about 3.10 percentage points, so the interval is about 46.90% to 53.10%.

A blank design effect is treated as 1. If the known population equals the completed sample, the finite population correction is zero because every member of that closed population was included.