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Confidence Interval Calculator (Mean)

Compute a confidence interval for a population mean from raw data or summary statistics, using the t-distribution — correct for the small samples the normal formula gets wrong.

Mean
49.0000
95% CI
35.5407 – 62.4593
Margin of error
± 13.4593
n = 5 · SD = 10.8397 · SE = 4.8477
t(4, 95%) = 2.776
Interval = mean ± t × SE

The Formula Behind the Result

  1. Paste raw measurements separated by commas, spaces, or newlines — the tool computes mean, standard deviation, and n for you.
  2. Or switch to summary mode and enter the mean, standard deviation, and sample size directly.
  3. The interval uses the t-distribution with n−1 degrees of freedom, which widens intervals honestly for small samples.
  4. Standard error and margin of error are reported separately so you can see what drives the width.
Formula & Technical Standard
CI = x̄ ± t(α/2, n−1) · s/√n. With s = sample SD (n−1 denominator), t critical values at 90/95/99% for the matching degrees of freedom.

Worked Example

Small usability study
Input: Task times 42, 58, 35, 61, 49 (n=5), 95%
Output: mean 49.0 s, 95% CI 34.6 to 63.4 s
With only 5 observations the t critical value is 2.78, not 1.96 — the interval is honest about how little data that is.
Summary-mode entry
Input: mean 182.4, SD 14.1, n = 120
Output: 95% CI 179.8 to 185.0
At n = 120 the t value is nearly 1.98; the interval is driven almost entirely by the standard error s/√n.

Common Questions

When should I use t instead of z (1.96)?

Whenever the standard deviation is estimated from the sample — which is almost always. The t-distribution accounts for that extra uncertainty; below n ≈ 30 the difference is material, and above a few hundred it becomes negligible.

What assumptions does this interval make?

Roughly independent observations and a distribution that is not extremely skewed (the Central Limit Theorem helps with larger n). For heavy-tailed data or tiny samples, consider reporting the median with a bootstrap interval instead.

Why is my interval so wide?

Width scales with s/√n: halving the width requires four times the data, and reducing measurement variability is often cheaper than collecting more. The tool separates SE and margin of error so you can see which lever matters.

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Confidence Interval Calculator (Proportion)

Build a confidence interval for a proportion from successes and trials — with the Wilson interval that stays accurate for small samples and 0% or 100% results.

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RMSE & MAE Calculator

Paste paired predictions and observations to get RMSE, MAE, MAPE, bias, and R² — the standard error metrics for evaluating forecasts, models, and measurements.

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Sample Size Calculator

Find the minimum sample size for a survey or experiment: one-proportion studies with a margin of error, or two-group comparisons powered at 80% with α = 0.05.