Margin of Error Calculator
Estimate the plus-or-minus sampling margin for a proportion with an explicit sample size, estimated proportion and selected confidence level.
Read the plus-or-minus number as a large-sample approximation
Worked example: with n = 1,000, an estimated proportion of 50% and 95% confidence, the standard error is √(0.5 × 0.5 ÷ 1000) ≈ 0.01581. Multiplying by 1.96 gives about 0.03099, so Lynara reports a margin of roughly ±3.10 percentage points.
The estimated proportion matters. With the same n = 1,000 and 95% confidence but p = 20%, the margin is about ±2.48 percentage points because p(1 − p) is smaller. This is why 50% is often used as a conservative planning value when the proportion is unknown.
Lynara also checks the expected success and failure counts. For n = 100 and p = 2%, the expected counts are 2 successes and 98 failures, so the page flags that the simple normal approximation is weak under its built-in 5-and-5 check. A different interval method may be more appropriate for small counts or proportions near 0% or 100%.
This number models sampling uncertainty under the stated simple-random-sample approximation. It does not include nonresponse, coverage error, question wording, weighting, clustering or other survey-design effects, so real-world total error can be larger.
Source: NIST Dataplot, Proportion Confidence Interval; NIST/SEMATECH guidance on large-sample normal approximations for proportions.
Privacy: calculator inputs are processed in your browser; Lynara does not need to upload these values to calculate the result.
Formula
margin of error = z* × √(p × (1 − p) ÷ n)
Add and subtract percentage points rather than a percent of the estimate
- With n = 400, estimated proportion 50% and 95% confidence, the standard error is 0.025.
- Multiplying by the normal critical value gives about 4.90 percentage points. The corresponding interval is about 45.10% to 54.90%.
What the result tells you
The margin is added directly to the estimated proportion in percentage-point units. It is not 4.90% of the 50% estimate.
Before you use the result
The normal approximation can perform poorly with few expected successes or failures. Nonresponse, weighting and nonrandom sampling are not fixed by increasing the number in this formula.
Useful for
- Statistics homework
- Survey proportion planning
- Sampling-error estimates
Questions
Why is 50% often used when the proportion is unknown?
For this formula, p = 50% produces the largest standard error and therefore a conservative margin estimate.
Does this include all survey error?
No. It covers only the modelled sampling margin under the entered assumptions.