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Survey Sample Confidence Interval Calculator

The Survey Sample Confidence Interval Calculator estimates a two-sided confidence interval for a survey proportion from the number of positive responses and total completed responses. It is suited to straightforward survey outcomes such as approval, awareness, adoption, preference, or any item coded as a binary yes/no response.

The calculator uses the normal approximation to the sample proportion and can optionally apply a finite population correction when the survey samples without replacement from a known population. It reports the observed proportion, margin of error, lower bound, and upper bound. The result is most useful as a quick precision check; small samples, extreme proportions, complex weighting, or clustered designs can require more specialized interval methods.

Survey interval inputs

Result
—
Approximate margin of error
Observed proportion—
Lower bound—
Upper bound—
Standard error—

1. Enter positive responses
Count responses in the category you are estimating, such as “yes,” “approved,” or “adopted.”

2. Enter completed responses
Use the total number of valid responses included in that proportion.

3. Select confidence
Choose the confidence level for the normal critical value.

4. Optionally enter population
Provide a finite population size only when sampling without replacement from a known population. Leave 0 for the large-population form.

5. Read the bounds and margin
The main result is the half-width margin of error. Lower and upper endpoints are shown separately in the result details.

Sample proportion p̂ = x / n Base SE = sqrt[p̂(1 − p̂) / n] FPC = sqrt[(N − n) / (N − 1)] (when N is entered) Margin of error = z × SE × FPC Approx. interval = p̂ ± margin of error

x is the number of positive responses, n is completed responses, N is population size when known, and z is the normal critical value for the selected confidence level. Displayed bounds are restricted to 0%–100%.

What the result means

The margin of error is the estimated distance from the observed survey proportion to each side of the normal-approximation confidence interval.

This interval does not account for survey weights, clustering, item nonresponse adjustments, or other design effects unless those are incorporated separately.

Given: 230 positive responses out of 400, 95% confidence, no finite population correction.

Calculation: p̂ = 230 / 400 = 0.575. SE = sqrt(0.575 × 0.425 / 400) ≈ 0.02472. Margin = 1.96 × 0.02472 ≈ 0.04846. Bounds = 0.575 ± 0.04846 = 0.5265 to 0.6235.

Result: Approximate margin of error = 4.85 percentage points, giving an interval of about 52.65% to 62.35%.

The sample estimate is 57.5%, while the interval summarizes sampling uncertainty under the model.

When should I enter a population size?

Enter it when the sample is drawn without replacement from a known finite population and the sampling fraction is not negligible. Otherwise leaving it at 0 is a reasonable large-population approximation.

Why can the interval differ from a Wilson interval?

This calculator uses the simple normal or Wald form. Wilson, exact, and other intervals use different constructions and often behave better with small samples or extreme proportions.

Does the confidence level describe the probability this one interval contains the true value?

In frequentist terms, the confidence level describes the long-run coverage of the interval procedure under its assumptions. It is not a posterior probability statement about the fixed parameter.

Can I use weighted survey results here?

Only with caution. A weighted proportion can have a different variance from an unweighted simple random sample, so using the raw response count may understate or overstate uncertainty.

What if positive responses are zero or equal the full sample?

The normal formula then gives a zero standard error, which is a known weakness of this method at the boundaries. An exact or Wilson interval is preferable in those cases.