Confidence Interval Calculator

Enter sample statistics to find the confidence interval.

Type

Sample mean / proportion

Std deviation (for mean only)

Sample size

Confidence level

Lower bound —

Upper bound —

Margin of error —

Standard error —

CI = x̄ ± Z × (s/√n) for means. CI = p̂ ± Z × √(p̂(1-p̂)/n) for proportions.

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How Confidence Intervals Work

A confidence interval gives a range of plausible values for a population parameter based on sample data. A 95% CI means that if you repeated the sampling 100 times, about 95 of those intervals would contain the true value.

The width of the interval depends on your confidence level, sample size, and variability. Larger samples and lower confidence levels produce narrower (more precise) intervals.

A 95% confidence interval means: if you repeated your sampling procedure many times, 95% of the resulting intervals would contain the true population mean. It does NOT mean "there's a 95% probability the true mean is in this range" — the true mean is fixed; the interval is random.

Formula for a mean with known standard deviation: CI = x̄ ± z × (σ/√n). For unknown σ (most real situations), use the t-distribution instead: CI = x̄ ± t × (s/√n), where t is the critical value from a t-table at n−1 degrees of freedom.

Example: 50 measurements with mean=45.2 and standard deviation=8.1. At 95% confidence, z=1.96. CI = 45.2 ± 1.96 × (8.1/√50) = 45.2 ± 2.24. The interval is [42.96, 47.44].

Narrowing the interval requires a larger sample (n appears under the square root — quadrupling n halves the margin of error) or accepting less confidence. Reducing from 95% to 90% shrinks the margin slightly (z drops from 1.96 to 1.645), but the precision gain is modest.