Confidence Interval Calculator

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Build a confidence interval for a mean from the sample mean, standard deviation and size. See the margin of error, lower and upper bounds and the z critical value. Free and live.

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Confidence interval –
Margin of error –
Lower bound –
Upper bound –
Z critical value –
Uses the z distribution, which assumes a large sample (n ≥ 30) or a known population standard deviation. The interval is mean ± z · σ ⁄ √n. For small samples, a t distribution is more appropriate.

About Confidence Interval Calculator

The confidence interval calculator builds an interval estimate for a population mean from three numbers: the sample mean, the standard deviation and the sample size n. Choose a confidence level — common presets or a custom percentage — and it reports the margin of error, the lower and upper bounds and the z critical value used.

The formula is the z-interval: mean ± z × σ ⁄ √n. It relies on the z distribution, which is appropriate for large samples (n ≥ 30) or when the population standard deviation is known; for small samples a t distribution is the better choice.

Confidence intervals put honest uncertainty around survey results, A/B test metrics and lab measurements — this tool makes the calculation free, live and transparent about its assumptions.

How to use Confidence Interval Calculator

  1. Enter the sample mean from your data.
  2. Enter the standard deviation and the sample size n.
  3. Pick a confidence level — 90%, 95%, 99% or a custom value.
  4. Read the confidence interval, the margin of error, the bounds and the z critical value.

Frequently asked questions

If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population mean. It quantifies estimation uncertainty, not the spread of individuals.

As z × σ ⁄ √n — the critical value times the standard error. At 95% confidence, z is 1.96, so the margin is roughly two standard errors either side of the sample mean.

When the sample is small (n below about 30) and the population standard deviation is unknown. The t distribution's wider tails give appropriately wider intervals; this calculator uses the z distribution.

Increase the sample size — the margin shrinks with √n, so quadrupling n halves the width — or accept a lower confidence level. Reducing measurement variability also helps.

Greater confidence demands a larger critical value (2.576 versus 1.96), stretching the bounds. You trade precision for certainty: wider intervals are more likely to capture the true mean.
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