Margin of Error Calculator
Estimate a normal-approximation margin of error for a sampled proportion.
How the result changes
Only observed sample proportion changes; all other inputs stay at your current values. The line illustrates the formula across the shown range. This is not a forecast of real-world outcomes.
About this calculation
Margin of Error Calculator: Estimate a normal-approximation margin of error for a sampled proportion. Enter observed sample proportion, sample size, normal critical value; the reported quantities are approximate margin of error (percentage points), lower approximate bound (%), upper approximate bound (%). The equations and examples below show how those inputs lead to the result and where the model's assumptions matter.
Formula
margin = z × sqrt(p(1-p)/n) × 100
Worked example
With Observed sample proportion: 0.5, Sample size: 400 observations, Normal critical value: 1.96, approximate margin of error is 4.9 percentage points.
Method and inputs
For an estimated proportion p, the plug-in standard error is the square root of p times one minus p divided by sample size. Multiply by a normal critical value z and by one hundred to express the margin in percentage points.
Worked example
For p = 0.5 from 400 observations, the standard error is 0.025. With z = 1.96, the margin is 4.9 percentage points and the simple interval is 45.1 to 54.9 percent.
At p = 0.5, the product p(1-p) is largest, producing the most conservative margin for a given n and z. At p = 0.1 with the same 400 observations, the plug-in margin is about 2.94 percentage points, though approximation checks matter more near a boundary.
How to read the result
This is a sampling-uncertainty approximation, not an allowance for nonresponse, selection bias or measurement mistakes. A z value of about 1.96 corresponds to a familiar two-sided 95-percent normal interval under its assumptions.
Doubling sample size does not halve the margin; the standard error declines with the square root of n. To cut a five-point margin roughly in half under the same assumptions, sample size must be about four times as large.
Assumptions and checks
Observations should come from a suitable random sample and be approximately independent. The normal approximation works poorly with very small samples or p near zero or one. Its raw bounds can exceed zero-to-one limits in those cases.
Entering 50 instead of 0.5 for a fifty-percent sample proportion is invalid. Do not interpret the resulting interval as a probability that a fixed population parameter moves around after sampling. More careful intervals, such as Wilson, may be preferable.
Common questions
What does the margin of error calculator calculate?
It reports an approximate proportion margin in percentage points and untruncated lower and upper normal bounds.
When should I recalculate or check another method?
Recalculate when sample size, observed proportion or confidence level changes. For survey reporting, review weighting, design effects and response patterns before publishing a simple random-sample margin.
Sources and references
Inputs
- Observed sample proportion
- Sample size (observations)
- Normal critical value