Confidence Interval Calculator

A confidence interval gives a range of likely values for something you can’t measure in full, such as the average height of every adult in a country, from a sample. It is the estimate from your sample plus and minus a margin of error. The margin grows with the spread of the data and the confidence you ask for, and shrinks as the sample gets bigger.

The formula

Mean (σ unknown): x̄ ± t* × s ÷ √n, with n − 1 degrees of freedom
Mean (σ known): x̄ ± z* × σ ÷ √n
Small population: multiply the standard error by √((N − n) ÷ (N − 1))
Proportion (Wald): p̂ ± z* × √(p̂(1 − p̂) ÷ n)
Proportion (Wilson): [p̂ + z²/(2n) ± z × √(p̂(1 − p̂)/n + z²/(4n²))] ÷ (1 + z²/n)
Proportion (Clopper–Pearson): from the beta distribution, Beta(α/2; x, n − x + 1) to Beta(1 − α/2; x + 1, n − x)
Two means (Welch): (x̄₁ − x̄₂) ± t* × √(s₁²/n₁ + s₂²/n₂), df = (s₁²/n₁ + s₂²/n₂)² ÷ [(s₁²/n₁)²/(n₁ − 1) + (s₂²/n₂)²/(n₂ − 1)]
Two proportions (Wald): (p̂₁ − p̂₂) ± z* × √(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂)

In words: the standard error is the standard deviation divided by the square root of the sample size. Multiply it by the critical value for your confidence level (t when the SD comes from your sample, z when it is known) to get the margin of error. The interval is the estimate minus the margin to the estimate plus the margin. For a two-sided 95% interval, t* leaves 2.5% in each tail; a one-sided 95% bound puts all 5% in one tail.

Worked example

A sample of 25 with a mean of 50 and a standard deviation of 10, at 95% confidence.

• Standard error = 10 ÷ √25 = 10 ÷ 5 = 2.
• Degrees of freedom = 25 − 1 = 24, so t* for 95% is 2.063899.
• Margin of error = 2.063899 × 2 = 4.127797.
• Interval = 50 − 4.127797 to 50 + 4.127797 = 45.872 to 54.128.
• If σ = 10 were known, z* = 1.959964 and the margin would be 1.959964 × 2 = 3.919928, giving 46.08 to 53.92.

A proportion: 40 out of 100 at 95%. p̂ = 0.4 and z = 1.959964, so z² = 3.841459. Wald: √(0.4 × 0.6 ÷ 100) = 0.048990, × 1.959964 = 0.096018, giving 30.40% to 49.60%. Wilson: the centre is (0.4 + 3.841459 ÷ 200) ÷ (1 + 3.841459 ÷ 100) = 0.419207 ÷ 1.038415 = 0.403699 and the half-width is 1.959964 ÷ 1.038415 × √(0.0024 + 0.000096) = 0.094298, giving 30.94% to 49.80%. Clopper–Pearson gives 30.33% to 50.28%.

Two means (Welch): 75 (s 10, n 30) against 70 (s 12, n 25). s₁²/n₁ = 3.333333 and s₂²/n₂ = 5.76, so the standard error is √9.093333 = 3.015515 and df = 9.093333² ÷ (3.333333²/29 + 5.76²/24) = 46.83. t* = 2.011928, margin = 6.067, interval −1.07 to 11.07. It includes 0, so the difference is not significant at 95%.

What 95% confidence means (and doesn’t)

• It is about the method. If you repeated the study many times and built an interval each time, about 95% of those intervals would catch the true value. The picture at the top of this page shows that: most bars cross the dashed line, and now and then one misses.
• It is not a 95% chance for your one interval. Once you have your numbers, the true value is either inside or not. You just don’t know which.
• It is not where 95% of the data are. That is a much wider range (roughly mean ± 2 SD). A confidence interval is about the average, not single values.
• It only covers random error. A biased sample (only volunteers, only one city) gives a tight interval around the wrong answer.

Critical t values

Which method to pick

• Mean: use t (the default). It assumes the data are roughly bell-shaped, or that the sample is big enough (about 30+) for the average to behave. For very skewed small samples, consider a bootstrap or a transformation.
• Proportion: Wilson for everyday use; Clopper–Pearson when you must never under-cover (it is conservative); Wald only for large samples away from 0% and 100%, as a rough check.
• Two means: Welch is the safe default because it doesn’t assume equal spreads. Pooled matches the classic Student’s t-test when the SDs are similar.
• Two proportions: Newcombe’s score method (from two Wilson intervals) behaves better than Wald for small groups or extreme rates.
• Paired data (before and after on the same people): work out each person’s difference first, then use “Mean (from raw data)” on those differences.

Critical values are computed from Student’s t distribution (incomplete beta function) and the normal distribution, and checked against published tables: t(0.975, 9) = 2.262157, t(0.975, 29) = 2.045230.