Confidence Interval Calculator
Calculate confidence intervals for population means (z and t distributions) and proportions. Get CI lower/upper bounds, margin of error, and required sample size.
CI Lower Bound
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CI Upper Bound —
Margin of Error —
Critical Value (z) —
Extended More scenarios, charts & detailed breakdown ▾
CI Lower
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CI Upper —
Margin of Error —
Professional Full parameters & maximum detail ▾
Current CI
CI Lower —
CI Upper —
CI Width —
Sample Size Planning
Required n for Target Width —
How to Use This Calculator
- Enter the Sample Mean, Standard Deviation, and Sample Size.
- Select the Confidence Level (90%, 95%, or 99%).
- The CI bounds and margin of error are calculated instantly.
- Use the For Proportion tab for survey/proportion data.
- The Professional tab shows required sample size for a target CI width.
Formula
CI = x̄ ± z × (σ/√n) | z = 1.645 (90%), 1.960 (95%), 2.576 (99%)
Proportion CI = p̂ ± z × √(p̂(1−p̂)/n)
Example
x̄=50, σ=10, n=36, 95%: SE=10/6=1.667, MoE=1.96×1.667≈3.27 → CI = [46.73, 53.27].
Frequently Asked Questions
- A confidence interval gives a range of values that likely contains the true population parameter. A 95% CI means: if we repeat the study many times, 95% of the computed intervals will contain the true value.
- Higher confidence level = wider interval. A 99% CI is wider than a 95% CI. The critical values are: z=1.645 (90%), z=1.960 (95%), z=2.576 (99%).
- Use t-distribution when the sample size is small (n < 30) and the population standard deviation is unknown. The t-distribution has heavier tails, producing wider intervals.
- CI = p̂ ± z × √(p̂(1−p̂)/n), where p̂ is the sample proportion. For p̂=0.6, n=100, 95%: MoE = 1.96 × √(0.6×0.4/100) ≈ 0.096.
- Required n = (z × σ / desired half-width)². For σ=10, 95% CI, half-width=2: n = (1.96×10/2)² = 96.04 → 97 samples.