Confidence interval calculator for a mean (t or z), a proportion (Wilson or Wald) or a difference of two means, with the margin of error and both bounds.
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The 95% confidence interval for the mean is 69.6594 to 75.1406 (72.4 ± 2.7406). Confidence describes the repeated-sampling method under its assumptions, not a probability for this particular interval.
For a mean, a confidence interval is a point estimate plus or minus a margin of error: a critical value times the standard error. Proportion intervals can be asymmetric around the sample proportion. For a mean the standard error is s/√n and the critical value comes from Student's t with n − 1 degrees of freedom, or from the normal distribution when σ is known. For a proportion the Wilson score interval is the default, with the Wald formula p̂ ± z√(p̂(1 − p̂)/n) offered for comparison.
Survey results, lab measurements and A/B tests are reported this way. The default sample of 36 with mean 72.4 and standard deviation 8.1 gives SE = 1.35, t = 2.030 on 35 df and a 95% interval of 69.66 to 75.14, or 72.4 ± 2.74.
The confidence level describes the procedure under its sampling and distribution assumptions, not a probability assigned to a fixed parameter in this one interval. Approximate methods such as Wilson or Wald need not achieve exactly the nominal coverage. The first formula is for a mean with unknown σ; the second gives Wilson proportion bounds. Two-sample methods require independent groups, and the pooled method also assumes equal variances.
A confidence interval for a mean is x̄ ± t × s ÷ √n, with t taken from n − 1 degrees of freedom. For 36 measurements with mean 72.4 and SD 8.1, the standard error is 8.1 ÷ 6 = 1.35 and t = 2.030, so the 95% interval is 72.4 ± 2.74, or 69.66 to 75.14.
Worked step by step:
Changing the confidence level changes only the critical value. At 90% the interval is 70.12 to 74.68, and at 99% it is 68.72 to 76.08.
For a proportion p̂ from n responses, the textbook margin of error is E = z × √(p̂(1 − p̂) ÷ n). If 540 of 1,000 people say yes, p̂ = 0.54 and the standard error is √(0.54 × 0.46 ÷ 1,000) = 0.01576.
The 95% margin is 1.96 × 0.01576 = 0.0309, about ±3.1 percentage points. Because n sits under a square root, quadrupling the sample to 4,000 only halves the margin, to ±1.5 points.
The formula above is the Wald interval. It covers the true proportion less often than stated when n is small or p̂ is near 0 or 1; Brown, Cai and DasGupta (2001) found its coverage can fall far below 95% even with hundreds of observations.
The Wilson score interval stays close to the stated level and never leaves the range 0 to 1 (NIST). For 540 of 1,000 it gives 0.509015 to 0.570679, close to Wald's 0.509110 to 0.570890.
The methods split apart at the extremes. With 0 successes in 20 trials, Wald gives the zero-width interval [0, 0], while Wilson gives 0 to 0.161. Seeing no successes does not show that the true rate is zero.
Wilson bounds are not centered on p̂, so the margin reported with them is half the interval's width, 0.030832 for 540 of 1,000. Quote the two bounds, since p̂ ± that margin misses them slightly.
Use z only when the population σ comes from outside the sample, as with a long-running process. A large sample alone does not make σ known. With σ = 8, the same mean and n = 36, the 95% z interval is 69.79 to 75.01, a margin of 2.613 (NIST).
The t critical value is larger for small samples and shrinks toward z as n grows: 2.262 at n = 10, 2.030 at n = 36 and 1.984 at n = 101, against 1.960 for z.
For two independent groups, the interval is (x̄₁ − x̄₂) ± t × SE. Welch's method uses SE = √(s₁²/n₁ + s₂²/n₂) with a fractional df, while the pooled method assumes the two populations have equal variances.
Take group 1 with mean 72.4, SD 8.1 and n = 36, and group 2 with mean 68.9, SD 9.4 and n = 40. The difference is 3.5, and the Welch SE is 2.0079 on 73.87 df.
With t = 1.9926, the 95% interval is 3.5 ± 4.0009, from −0.50 to 7.50. Pooled variances give −0.53 to 7.53 on 74 df.
The interval includes 0, so the difference is not significant at the 5% level. A 95% interval excludes 0 exactly when the matching two-sided test, Welch or pooled, rejects at α = 0.05.
The interval estimates mean 1 minus mean 2, so swapping the groups flips the signs. Paired data, such as before-and-after scores for the same people, need an interval for the within-pair differences instead.
The 95% describes the method: across many samples, 95% of intervals built this way contain the true mean. Any one interval, such as 69.66 to 75.14, either contains it or does not.
The interval is not a range for individual values either. With an SD of 8.1, single measurements spread far wider than ±2.74; covering them needs a prediction or tolerance interval.
These intervals assume a random sample of independent observations, and two-sample intervals need independent groups.
The t interval also assumes a roughly normal population. That matters less as n grows, though strong skew or outliers still distort a small sample.
| Scenario | Lower bound | Upper bound |
|---|---|---|
| Mean, sample SD, default inputs | 69.659354 | 75.140646 |
| Mean, known σ = 8 | 69.786715 | 75.013285 |
| Proportion 540/1000, Wilson | 0.509015 | 0.570679 |
| Proportion 0/20, Wilson | 0.000000 | 0.161125 |
| Default difference of means, Welch | -0.500864 | 7.500864 |
The first two rows use x̄ = 72.4 and n = 36, with s = 8.1 or a known σ = 8, and the last row is the two-group comparison above. All are 95% intervals.
To test a specific value, such as whether the mean differs from 70, use the t-test calculator. To choose n for a target margin of error, use the sample size calculator.
Checked against: t₀.₉₇₅,₃₅ = 2.030 (t table); bisection on the A&S 26.7.3 closed form gives 2.0301079283; margin = t·8.1/6
Checked against: z₀.₉₇₅ = 1.959964; margin = z·8/6
Checked against: Independent calculation
Checked against: Independent calculation
The method captures the true value in 95% of repeated samples. For the default data the interval is 69.66 to 75.14, but there is not a 95% probability that the true mean lies in that particular range: the true mean is fixed, and this interval either contains it or not. Nor does the interval hold 95% of individual values, which spread far wider (SD 8.1); that needs a prediction or tolerance interval.
Multiply the critical value by the standard error. For a mean, E = t × s/√n: with s = 8.1 and n = 36, E = 2.030 × 1.35 = 2.74. For a proportion, E ≈ z√(p̂(1 − p̂)/n): 54% of 1,000 respondents gives 1.96 × 0.0158 ≈ 0.031, or ±3.1 percentage points. Because n sits under a square root, quadrupling the sample size halves the margin.
Only when the population standard deviation σ is known, which is rare outside textbook problems and long-running process data. With σ estimated from the sample, use t. Its 95% critical value is larger for small samples, 2.262 for n = 10 against 1.960 for z, and approaches z as n grows: 2.030 at n = 36 and 1.984 at n = 101.
The Wald interval p̂ ± z·SE covers the true proportion less often than stated when n is small or p̂ is near 0 or 1; Brown, Cai and DasGupta (2001) found its coverage can fall far below 95% even with hundreds of observations. Wilson stays close to the stated level and never leaves the range 0 to 1. With 0 successes in 20 trials Wald gives the zero-width interval [0, 0], while Wilson gives 0 to 0.161.
Yes, for the matching test: a 95% interval for a difference in means excludes 0 exactly when a two-sided test at α = 0.05 using the same method (Welch or pooled) rejects no difference. The default two-group summaries give 3.5 ± 4.0, from −0.5 to 7.5, which includes 0, so the difference is not significant at 5%. The interval also shows how large the effect could plausibly be, which a p-value does not.
This calculator includes 7 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.