Confidence Interval Calculator
Enter sample mean, standard deviation, size and confidence level to get the confidence interval, margin of error and standard error. For statistical analysis and surveys.
Input Data
Results
At a glance:A Confidence Interval gives an estimated range for a population mean at a given confidence level (e.g. 95%). For large samples the sampling distribution of the mean is approximately normal; the standard error is SE = s/√n. The CI is x̄ ± z·SE, where z is the standard-normal critical value for the level (90%→1.645, 95%→1.960, 99%→2.576). '95% confidence' means that if you repeatedly sampled and computed intervals, about 95% would contain the true value.
Formula
Standard error: SE = s ÷ √n
Margin of error: E = z × SE
Confidence interval: CI = x̄ ± E, i.e. [x̄ − E, x̄ + E]
Example: x̄=500, s=80, n=100, 95% → SE=8, E≈15.68, CI=[484.32, 515.68]
$$\text{CI} = \bar{x} \pm z \cdot \frac{s}{\sqrt{n}}$$How to Use
- Enter the sample mean x̄.
- Enter the sample standard deviation s and size n.
- Choose the confidence level.
- The tool returns SE, margin of error and the CI bounds.
Case Studies
Survey mean estimate
For x̄=500, s=80, n=100 at 95%, the 95% CI is [484.32, 515.68].
FAQ
Does 95% mean the true mean has a 95% chance to be inside?
No — the interval is random, the parameter is fixed. It means 95% of such intervals from repeated sampling would contain the true value.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.