How to use the Confidence Interval
- Pick 'Mean' for measured quantities, or 'Proportion' for counts of successes out of a total.
- Enter the sample statistics: mean and standard deviation, or successes and sample size.
- Enter the sample size — this is what shrinks the interval.
- Choose the confidence level; 95% is the near-universal default.
- Report the interval, not just the point estimate, whenever you quote a survey or experimental result.
How the calculation works
A confidence interval expresses the precision of an estimate. The procedure, not the individual interval, carries the guarantee: if you repeated the sampling many times and built a 95% interval each time, about 95% of those intervals would contain the true parameter. Any single interval either contains it or does not, which is why 'there is a 95% chance the true value is in this range' is a Bayesian statement that this frequentist interval does not actually make.
The width is the margin of error, z × standard error, and the standard error falls with the square root of n. That square root governs all survey economics: quadrupling the sample halves the margin. It also explains why national polls of 1,000 people carry a margin near ±3 points regardless of whether the country has 5 million or 300 million people — population size barely enters unless the sample is a large fraction of it.
For proportions the standard error is √(p(1 − p)/n), which peaks at p = 0.5 and shrinks as the proportion approaches either extreme. The normal approximation used here needs roughly ten successes and ten failures to behave; for very rare events, exact Clopper–Pearson intervals are the right tool and this one will be too narrow and can even run past 0% or 100%.
Mean: x̄ ± z × s/√n. Proportion: p̂ ± z × √(p̂(1 − p̂)/n). z = 1.645, 1.960, 2.326, 2.576 for 90%, 95%, 98%, 99%.Source: Neyman, 'Outline of a Theory of Statistical Estimation', Phil. Trans. R. Soc. A 236 (1937); NIST/SEMATECH e-Handbook, section 1.3.5.2.
Worked example
A poll of 1,000 voters finds 480 support a measure. What is the 95% confidence interval?
- p̂ = 480 ÷ 1000 = 0.480.
- Standard error = √(0.48 × 0.52 ÷ 1000) = 0.0158.
- Margin of error = 1.96 × 0.0158 = 0.0310, or 3.1 percentage points.
- Interval = 48.0% ± 3.1% = 44.9% to 51.1%.
Support is 48% with a 95% interval of 44.9%–51.1%. Because the interval spans 50%, the poll cannot establish that the measure is behind.
Frequently asked questions
Why is 95% the default?+
Convention, dating to Fisher's suggestion that two standard deviations is a convenient threshold. Nothing mathematical distinguishes it from 94% or 96%.
Does a wider interval mean worse data?+
It means less precision, which comes from a small sample, high variability, or a higher confidence level. Higher confidence always costs width.
How do I halve my margin of error?+
Quadruple the sample size. The margin scales with 1/√n, so there is no cheaper route.
When should I use a t interval instead?+
For means from samples under about 30 where the population standard deviation is unknown. The t multiplier is larger and the interval correspondingly wider.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.