Math & Statistics
Proportion Confidence Interval Calculator
Calculate observed proportion, failures, design-adjusted Wald standard error, critical margin, clipped lower and upper bounds, interval width, and comparison difference with a step-by-step derivation and method-specific cautions.
Observed sample proportion (decimal)-
Wald standard error-
Critical-value margin-
Continuity correction in proportion units-
Critical margin plus continuity correction-
Continuity-adjusted lower bound clipped at zero-
Continuity-adjusted upper bound clipped at one-
Entered comparison proportion (decimal)-
Observed minus comparison proportion-
Clipped interval width-
Observed non-successes-
Decision view
Observed composition and proportion risk band
Observed composition and proportion risk bandSuccesses and failures form the observed sample while the clipped Wald interval and comparison proportion remain explicit.
Exact scenario comparisonObserved successes changes while all other entered assumptions remain constant.
| Observed successes | Observed sample proportion (decimal) | Wald standard error | Critical-value margin | Continuity correction in proportion units | Critical margin plus continuity correction | Continuity-adjusted lower bound clipped at zero | Continuity-adjusted upper bound clipped at one | Entered comparison proportion (decimal) | Observed minus comparison proportion | Clipped interval width | Observed non-successes |
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How to use Proportion Confidence Interval Calculator
- Enter success count and sample size from the same observation set.
- Confirm successes do not exceed the sample size.
- Choose a critical value and design effect appropriate to the analytical design.
- Check whether a Wald interval is suitable before relying on the displayed bounds.
Calculator guide
Understanding Proportion Confidence Interval Calculator
A proportion interval begins with an observed success share and a sampling-variation model. This page makes the Wald standard error, design effect, clipping, failures, and comparison proportion visible instead of returning only two bounds.
Calculation method
How the calculation works
Calculate a transparent normal-approximation proportion interval from observed successes, sample size, critical value, and design effect, then add the entered continuity correction count divided by sample size to each side of the margin. In the Proportion Confidence Interval Calculator, the live scenario varies observed successes and tracks observed sample proportion (decimal) while the remaining results preserve the reconciliation path. Divide observed successes by sample size. Multiply p by one minus p, divide by n, multiply by design effect, and take the square root for the Wald SE. Apply the entered critical value and clip arithmetic bounds to zero and one.
Detailed calculation process
Build the observed-proportion risk band
The default sample contains 132 successes and 68 failures, yielding an observed proportion of 0.66.
General formula: p = x/n; SE = sqrt[p(1-p)DEFF/n]; interval = clip(p +/- c x SE, 0, 1) The observed proportion determines both the center and binomial variance term. Design effect scales variance, the critical value sets the margin, and clipping keeps displayed proportions inside logical bounds.
What each symbol means
Worked substitution with the default inputs
The defaults produce an observed 66% share and a clipped Wald interval of approximately 59.435% to 72.565%.
Interval audit
Read composition and uncertainty together
The observed sample and the uncertainty model are separate layers.
Worked situations
Practical examples
- 132 successes in 200 observations produce p = 0.66.
- With design effect one, SE is about 0.0335.
- The default 60% comparison lies inside the calculated interval.
Better inputs
Useful tips
- Report counts with the proportion because identical percentages can come from very different sample sizes.
- Use Wilson, exact, or survey-specific methods when the Wald approximation is weak.
- Keep design-based weights and clustering in the variance method rather than treating them as an afterthought.
Before relying on the result
Limitations and common mistakes
- Wald intervals can have poor coverage for small n or proportions near zero or one.
- Clipping prevents impossible displayed bounds but does not repair an unsuitable approximation.
- Nonresponse, measurement error, coverage error, dependence, and weighting require separate treatment.
Reference
Key terms
- Sample proportion
- Success count divided by total observations.
- Wald interval
- Normal-approximation interval centered on the observed proportion.
- Design effect
- Variance multiplier relative to a stated simple-sampling model.
- Clipping
- Restricting displayed bounds to the logical 0-to-1 range.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why is the interval centered on the observed proportion?
That is the defining structure of this transparent Wald calculation.
What if successes exceed sample size?
The inputs are inconsistent and should be corrected before any interpretation.
Why clip bounds to zero and one?
Proportions cannot be outside that range, though clipping does not improve the approximation's statistical coverage.
When should I use Wilson instead?
Wilson is often preferred for smaller samples or proportions away from the middle; choose based on the reporting standard and analysis plan.