Confidence Interval Calculator

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Sample Mean (x̄)

Standard Deviation (σ or s)

Sample Size (n)

Confidence Level90%95%99%

Standard Deviation TypePopulation (σ) - Use z-distributionSample (s) - Use t-distribution

Calculate Confidence IntervalClear

Inputs and method

This tool calculates a two-sided confidence interval for one population mean from summary data. It does not calculate intervals for proportions, individual predictions, correlations, or differences between two means.

1. Enter the sample mean, using the original measurement units.
2. Enter a positive SD and an integer sample size of at least two.
3. Choose 90%, 95%, or 99% confidence.
4. Specify whether SD is a known population value or a sample estimate.
5. Calculate, then interpret the interval alongside the design and assumptions.

Unknown population SD uses a t critical value with df = n − 1. Known population SD uses a standard normal critical value. A large sample does not make an estimated SD known.

Reproduce the example

Enter mean 0.45, SD 0.08, n = 25, confidence 95%, and unknown population SD.

$SE=0.08/\sqrt{25}=0.016$

With df = 24, t* ≈ 2.0639. The margin is about 0.03302, giving an interval about [0.41698, 0.48302] in the original units.

Changing only the confidence level to 99% widens the interval. Greater coverage comes with less precision; it does not improve the estimate itself.

Interpret confidence correctly

The confidence level refers to repeated-sampling coverage of the method under its assumptions. It is not the probability that the fixed mean lies inside this one observed interval, and it is not the percentage of individual observations in the interval.

A matching two-sided test at α = 0.05 rejects a null mean outside a 95% interval. An interval including the null does not demonstrate equality or practical equivalence.

Check design and distribution

The standard mean interval assumes suitable independent observations. Exact small-sample t inference uses a normal population model. Inspect strong skew and extreme influence, and consider design-aware methods for clustered, paired, or serial data.

Sampling bias and measurement errors are not included automatically in the interval. A narrow interval from biased data can still be misleading. State the population, method, units, and limitations when reporting the endpoints.

The known-SD mode uses rounded standard normal cutoffs 1.645, 1.960, and 2.576. Differences in the last digits from software using full-precision cutoffs can reflect rounding.

Related learning

Read confidence intervals for the derivation, proportion-method limitations, and practice. Use the sample size calculator for approximate planning and the t-table to inspect critical values.