Math & Stats
Confidence Interval Calculator
Calculate the confidence interval from sample statistics. Get the margin of error, lower and upper bounds, and a plain-language interpretation.
Enter sample statistics and click Calculate
Confidence Interval Formula
CI = x̄ ± z* × (σ / √n)
Margin of Error: E = z* × (σ / √n)
Lower Bound: x̄ − E
Upper Bound: x̄ + E
z* values:
90% → 1.645
95% → 1.960
99% → 2.576FAQ
Frequently asked questions about confidence intervals
What is a confidence interval?
A confidence interval is a range of values that is likely to contain the true population parameter. A 95% CI means that if you repeated the experiment many times, about 95% of the calculated intervals would contain the true mean.
What's the difference between 90%, 95%, and 99% confidence levels?
Higher confidence levels produce wider intervals. A 99% CI is wider than 95% which is wider than 90%. You gain more certainty but lose precision. 95% is the most commonly used level in research.
When should I use z-score vs t-score?
Use z-score when sample size is large (n ≥ 30) or population standard deviation is known. Use t-score for small samples (n < 30) with unknown population standard deviation. This calculator uses z for n≥30 and t-approximation for smaller samples.
What is margin of error?
Margin of error is the half-width of the confidence interval: E = z*(σ/√n). It represents the maximum expected difference between the sample statistic and the true population parameter at the given confidence level.
How does sample size affect the confidence interval?
Larger sample sizes produce narrower (more precise) confidence intervals because the standard error (σ/√n) decreases as n increases. Quadrupling the sample size halves the margin of error.
Can a confidence interval include negative values?
Yes. If your sample mean minus the margin of error is negative, the lower bound will be negative. This is valid and simply means the true population mean could potentially be negative based on your data.