Math & Statistics
Survey Margin of Error Calculator
Calculate simple-random-sample SE, finite-population correction, design-adjusted SE, percentage-point margin, interval around the expected proportion, and comparison gap. Review what sampling margin includes and what it omits.
Margin of error (%)-
Expected proportion as decimal-
Simple-random-sample standard error-
Finite population correction-
Design and finite-population adjusted SE-
Margin minus entered comparison-
Expected proportion minus margin-
Expected proportion plus margin-
Decision view
Margin-of-error sensitivity to completed sample size
Margin-of-error sensitivity to completed sample sizeA descending risk curve shows how the current adjusted margin changes as sample size increases under the same assumptions.
Exact scenario comparisonCompleted sample size changes while all other entered assumptions remain constant.
| Completed sample size | Margin of error (%) | Expected proportion as decimal | Simple-random-sample standard error | Finite population correction | Design and finite-population adjusted SE | Margin minus entered comparison | Expected proportion minus margin | Expected proportion plus margin |
|---|
How to use Survey Margin of Error Calculator
- Enter the expected proportion and completed sample size.
- Enter the finite population only when a defined sampling frame makes the correction appropriate.
- Apply a documented design effect and critical value.
- Use the sensitivity curve to see diminishing precision gains as completed sample increases.
Calculator guide
Understanding Survey Margin of Error Calculator
Survey margin of error depends on expected proportion, completed sample size, critical value, design effect, and the population fraction sampled. This page shows each adjustment and a sample-size sensitivity curve rather than presenting one isolated percentage.
Calculation method
How the calculation works
Combine binomial standard error, square-root design effect, finite-population correction, and an entered critical value into a percentage-point margin. In the Survey Margin of Error Calculator, the live scenario varies completed sample size and tracks margin of error (%) while the remaining results preserve the reconciliation path. Convert the expected percentage to a proportion, compute binomial SE, multiply by the square root of design effect and the finite-population correction, then multiply by the critical value and 100 for a percentage-point margin.
Detailed calculation process
Adjust binomial sampling margin for design and population
The default calculation begins at maximum binomial variance p = 0.5 and applies both a 1.2 design effect and a finite-population correction.
General formula: MOE% = c x sqrt[p(1-p)/n] x sqrt(DEFF) x sqrt[(N-n)/(N-1)] x 100 The first square root is simple binomial SE. Design effect adjusts variance, finite-population correction recognizes sampling without replacement, and c sets the chosen critical-width convention.
What each symbol means
Worked substitution with the default inputs
The defaults produce an adjusted sampling margin of approximately 4.272 percentage points around the expected 50% proportion.
Error-budget check
Keep sampling margin in its proper lane
A narrow sampling margin can coexist with large nonsampling errors.
Worked situations
Practical examples
- At 50%, n = 600, DEFF = 1.2, and N = 12,000, margin is about 4.272 points.
- Increasing n lowers margin approximately with the inverse square root until the finite correction becomes material.
- Changing expected p away from 50% lowers the model's p(1-p) variance term.
Better inputs
Useful tips
- Use completed responses, not invitations, for n.
- Do not apply finite correction merely because a population number is available; the sampling mechanism must support it.
- Budget separately for nonresponse, coverage, measurement, weighting, and processing error.
Before relying on the result
Limitations and common mistakes
- Sampling margin does not cover nonresponse, undercoverage, wording, mode, weighting, fraud, or measurement bias.
- The formula assumes a proportion estimator and design effect represented by one scalar.
- A comparison margin is a planning reference, not a guarantee of achieved survey quality.
Reference
Key terms
- Margin of error
- Critical-value multiple of adjusted sampling standard error, in percentage points.
- Finite population correction
- Variance reduction for sampling a substantial share without replacement.
- Design effect
- Variance multiplier relative to a reference simple design.
- Completed sample
- Usable observations included in the estimate.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why is 50% often used for planning?
Because p(1-p) is largest at 0.5, producing the largest binomial sampling margin for fixed n.
Should invitations be used as sample size?
No. Use completed usable responses; invitations belong in response-rate planning.
When does finite population correction matter?
It becomes more material as the sampled share of a well-defined finite population grows.
Does 4% margin mean every survey error is within 4 points?
No. It describes only modeled sampling variation, not the many nonsampling error sources.