Free confidence interval calculator
Calculate confidence intervals for a population mean or proportion — enter your sample data to see the interval and margin of error, updated live, as you type.
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CI for a mean. Uses z-scores: 1.645/1.96/2.576. Assumes normal distribution.
Results are estimates. Consult a professional.
How the confidence interval calculator works
A confidence interval (CI) gives a range of plausible values for an unknown population parameter based on sample data. A 95% CI means that if you repeated the study many times, 95% of the intervals you construct would contain the true population value. This calculator computes CI for a population mean using a known or estimated standard deviation.
Worked example: estimating average test score
A school samples 100 students. The sample mean score is 50 with a known standard deviation of 10. Construct a 95% confidence interval for the true mean score.
Critical z-values by confidence level
The critical z-value (z*) is the multiplier that sets how wide your interval is. Higher confidence requires a wider interval, captured by a larger z*.
| Confidence level | z* (critical value) | Interpretation |
|---|---|---|
| 80% | 1.282 | 1 in 5 intervals miss the true value |
| 90% | 1.645 | 1 in 10 intervals miss the true value |
| 95% | 1.960 | 1 in 20 intervals miss the true value |
| 99% | 2.576 | 1 in 100 intervals miss the true value |
| 99.9% | 3.291 | 1 in 1000 intervals miss the true value |
Source: Standard normal distribution. Use t-critical values instead when σ is unknown and n < 30.
Tips for working with confidence intervals
Confidence intervals are more informative than a single point estimate because they communicate both the value and the uncertainty around it.
- Increase n to shrink the interval — the margin of error shrinks with √n; quadrupling your sample size halves the width of the CI.
- Higher confidence = wider interval — moving from 95% to 99% confidence makes the interval wider; there is a trade-off between certainty and precision.
- Non-overlapping CIs suggest significance — if two confidence intervals for independent groups do not overlap, the difference is almost certainly statistically significant.
- Use t-distribution for small samples — when n < 30 and σ is unknown, use the t critical value at n−1 degrees of freedom rather than z = 1.96.
- CI is not a probability about a fixed parameter — once constructed, the interval either contains the true value or it doesn't; the 95% refers to the long-run procedure, not this specific interval.
Accuracy and limitations
This calculator assumes the sample mean is approximately normally distributed, which holds when n ≥ 30 by the central limit theorem, or when the population is itself normal. The formula also requires that observations are independent and drawn at random from the population. For proportions, use the Wilson score interval rather than the normal approximation when p is near 0 or 1, or n is small. Cluster sampling, stratified designs, and weighted surveys require modified formulas not covered here.
Key terms
About this calculator
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