Confidence Interval Calculator for a Mean
A confidence interval turns a single sample average into a defensible range: "we're 95% confident the true mean falls between here and here." It answers the question every reviewer asks — not just what did we measure, but how sure are we. Enter your summary statistics and the tool picks the right multiplier (z or t) for you.
Build the Interval
Which Confidence Interval Do You Need?
"Confidence interval" isn't one formula — it depends on what you're estimating and what you know. Pick the wrong row and your interval is meaningless.
CI for a mean vs proportion vs difference
| Mean (σ unknown)Best | Mean (σ known) | Proportion | Diff. of means |
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This calculator covers the highlighted row — the mean with unknown σ, which is what you'll actually face in the field.
Worked Example: Average Assembly Time
An engineer times 24 assemblies: average 7.8 minutes, sample SD 1.5. For 95% confidence she uses the t-multiplier with df = 23, which is 2.069. The standard error is 1.5 / √24 = 0.306, so the margin of error is 2.069 × 0.306 = 0.633. The interval is 7.8 ± 0.633 → [7.17, 8.43] minutes. Note the t (2.069) is larger than the z she'd wrongly use (1.960) — ignoring that would understate her uncertainty.
The Margin of Error, Explained Simply
Every confidence interval is point estimate ± (critical value × standard error). The standard error is how much sample means bounce around (σ / √n); the critical value is how many of those bounce-units you need to cover your chosen confidence level (1.96 for 95% z, a bit more for t). Multiply them and you get the margin of error — the half-width. Double it for the full width. The only dial you typically control is n: quadruple it to halve the width.
Frequently Asked Questions
What does a 95% confidence interval actually mean?
It means 95% of intervals built this way across repeated samples would capture the true mean. Your single interval either contains it or doesn't — you just can't tell which.
Should I use z or t for a mean?
Use t (df = n − 1) whenever you estimate σ from the sample, which is almost always. Use z only with a genuinely known population σ or a very large n. This tool auto-selects.
What is the margin of error?
The half-width: critical value × standard error. It's how far the interval reaches on either side of the sample mean; double it for the full width.
Why does a larger sample shrink the interval?
Standard error is proportional to 1/√n, so width shrinks with the square root of n. Halving the width requires four times the observations.