Confidence Interval Calculator
Adjust the inputs below. Results update as you type.
How it works
For means: CI = x̄ ± z × (σ/√n). Use z=1.96 for 95% confidence, 2.576 for 99%. When σ is unknown and n < 30, use the t-distribution instead. For proportions: CI = p̂ ± z × √(p̂(1−p̂)/n). A 95% CI means if the experiment were repeated 100 times, roughly 95 intervals would contain the true parameter—not that there's a 95% chance the true value is in this interval.
Input guidance
- Check units and decimal placement before running calculations.
- For percentage and ratio tasks, define the baseline value clearly.
- Use rounded output carefully and keep full precision for intermediate steps.
The formula
CI = sample mean ± (z or t) × (standard deviation ÷ √n). The margin widens with higher confidence and narrows with larger samples.
Worked example
A mean of 50, SD 10, n = 100, at 95% confidence: 50 ± 1.96 × (10 ÷ 10) = 50 ± 1.96, so the interval is about 48.0 to 52.0.
More examples to test
- Baseline example: solve with clean numbers first to verify formula direction.
- Edge-case example: test near-zero or boundary values to avoid interpretation errors.
How to interpret results
Use outputs as quick checks, then verify with step-by-step work for graded or high-stakes use.
When this can be inaccurate
Errors usually come from unit mismatch, wrong baseline assumptions, or rounding too early in multistep problems.
Change history
- July 2026: Quality-reviewed for publication with formula checks and explanatory copy updates.
Site-wide corrections also appear on the corrections log.
Build math and logic skills with guided lessons
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Frequently asked questions
What does a 95% confidence interval mean?+
If you repeated the sampling many times, about 95% of the intervals would contain the true population value. It is a measure of estimate reliability.
Why does a bigger sample help?+
The margin shrinks with the square root of n, so larger samples give narrower, more precise intervals.
z or t distribution?+
Use z for large samples or known population SD; use the t distribution for small samples with an estimated SD.
Why do my classroom and calculator results differ?+
Differences usually come from rounding conventions, order-of-operations handling, or different baseline assumptions.
What is the safest way to validate an answer?+
Recompute with units shown and check against a second method or manual arithmetic for critical problems.