Confidence Interval Calculator
Calculate confidence intervals for population means and proportions.
A confidence interval calculator determines the range within which a population parameter is likely to fall at a given confidence level.
Examples
Survey result
Small sample
Large sample
Frequently Asked Questions
What is a confidence interval?
How does sample size affect the interval?
When should I use a t-distribution instead?
Quick Tips
- •Double-check your inputs — small errors lead to incorrect results.
- •Use a t-distribution instead when your sample size is below 30 and the population standard deviation is unknown.
- •Increasing the sample size narrows the confidence interval for the same confidence level.
A confidence interval calculator determines the range within which a population parameter is likely to fall at a given confidence level.
How to Use This Calculator
Enter the sample mean, standard deviation, sample size, and desired confidence level. The calculator will compute the confidence interval bounds and margin of error using the z-distribution.
Understanding the Formula
CI = x̄ ± Z × (σ / √n), where Z depends on the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
Examples
Survey result
Mean 50, std dev 10, n=100, 95% CI: 50 ± 1.96 × (10/10) = (48.04, 51.96)
Small sample
Mean 75, std dev 5, n=25, 99% CI: 75 ± 2.576 × (5/5) = (72.42, 77.58)
Large sample
Mean 100, std dev 15, n=400, 90% CI: 100 ± 1.645 × 0.75 = (98.77, 101.23)
Frequently Asked Questions
What is a confidence interval?
A confidence interval gives a range of values that is likely to contain the true population parameter. A 95% CI means that if we repeated the sampling many times, 95% of intervals would contain the true value.
How does sample size affect the interval?
Larger sample sizes produce narrower confidence intervals because the standard error (σ/√n) decreases as n increases.
When should I use a t-distribution instead?
Use a t-distribution when the sample size is small (typically n < 30) and the population standard deviation is unknown. This calculator uses the z-distribution, suitable for larger samples.
Assumptions & Limitations
- Assumes a standard normal (z) distribution, which is most accurate for large samples.
- Assumes the population standard deviation is known or well-estimated.
- Assumes exact input values; rounding in inputs propagates to results.