Confidence Interval Calculator
Calculate the confidence interval for a population mean from a sample mean, standard deviation and sample size.
95% CI
You can be 95% confident the true population mean lies between 47.521 and 52.479 — the teal band is that interval, sized against a wider scale so you can see how wide (or narrow) it really is.
- Margin of Error± 2.4792
About the Confidence Interval Calculator
A confidence interval takes a single sample mean and turns it into a range that's likely to contain the true population mean — the number you'd get if you could somehow measure every single member of the population instead of just a sample. Almost no real-world measurement is exact, so this range is often the more honest, useful answer than a single point estimate on its own.
It's a core tool in research, quality control, and any A/B test or survey analysis — anywhere someone has measured a sample (average test scores, average product weight off an assembly line, average response to a survey question) and needs to state how much uncertainty surrounds that average.
The width of the interval is a direct trade-off with confidence: a wider interval lets you be more confident the true value falls inside it, while a narrower interval is more precise but riskier — it's more likely to miss the true value entirely.
How it’s calculated
The interval is built as mean ± z × (σ / √n): take the sample mean, then add and subtract a margin built from the z-score for your confidence level, the sample's standard deviation, and the square root of the sample size.
Standard deviation measures how spread out the individual data points are — more spread means more uncertainty about where the true mean sits, and the formula reflects that directly. Sample size works in the opposite direction: as √n grows, the margin shrinks, because a larger sample pins the average down more precisely.
A 95% confidence level means that if you repeated this same sampling process many times, about 95% of the resulting intervals would contain the true population mean — it's a statement about the reliability of the method, not a 95% probability attached to this one specific interval.
Frequently asked questions
What does a 95% confidence interval actually mean?
It means that if the same sampling process were repeated many times, about 95% of the resulting intervals would contain the true population mean. It doesn't mean there's a 95% chance the true mean falls in this particular interval — the true mean is fixed, only the interval varies.
Why does a larger sample size narrow the confidence interval?
The margin of error depends on the sample size through √n in the denominator — a bigger sample gives a more precise estimate of the true mean, which shrinks the range needed to be confident it's captured.
Should I use 90%, 95%, or 99% confidence?
95% is the standard default across most research and business contexts. Use 99% when the cost of being wrong is high and you want more certainty (at the cost of a wider interval); 90% when a rougher estimate is acceptable and you'd rather have a tighter range.
What's the difference between this and the sample size calculator?
This calculator starts from a sample you've already collected and tells you the confidence interval around its mean. The sample size calculator works in the opposite direction — it tells you how large a sample you'd need to collect to hit a target margin of error before you start.
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