Calculate confidence intervals for means, proportions, and differences with interactive visualization.
Enter your parameters and click "Calculate Confidence Interval" to see results.
A confidence interval (CI) is a range of values that likely contains the true population parameter with a specified level of confidence. For example, a 95% CI means we're 95% confident the true value lies within this range.
Point estimates alone are insufficient: A single sample statistic (like a mean) doesn't tell us about uncertainty.
CI provides a range: Shows precision of our estimate and accounts for sampling variability.
Better decision making: Helps assess practical significance beyond just statistical significance.
Correct: "We are 95% confident that the true population mean lies between 45 and 55."
Incorrect: "There's a 95% probability the true mean is in this interval." (The true mean is fixed, not random)
Width matters: Narrower CI = more precise estimate. Wider CI = more uncertainty.
| Confidence Level | Critical Value (Z) | Use Case |
|---|---|---|
| 90% | 1.645 | Quick estimates, exploratory analysis |
| 95% | 1.96 | Standard for most research |
| 99% | 2.576 | High-stakes decisions, critical applications |
Formula: CI = x̄ ± t × (s/√n)
Where: t = t-critical value for (n-1) degrees of freedom
Example: Mean = 50, s = 10, n = 30, 95% CI → [46.24, 53.76]
Formula: CI = p̂ ± Z × √(p̂(1-p̂)/n)
Where: p̂ = x/n (sample proportion)
Example: x = 45, n = 100, 95% CI → [35.3%, 54.7%]
Formula: CI = (x̄₁ - x̄₂) ± t × SE
Where: SE = √(s₁²/n₁ + s₂²/n₂)
Use: Comparing two groups (e.g., treatment vs control)
Formula: CI = (p̂₁ - p̂₂) ± Z × SE
Where: SE = √(p̂₁(1-p̂₁)/n₁ + p̂₂(1-p̂₂)/n₂)
Use: A/B testing, comparing success rates
Every coffee helps keep the servers running. Every book sale funds the next tool I'm dreaming up. You're not just supporting a site — you're helping me build what developers actually need.
In repeated sampling, 95% of such intervals would capture the true parameter. It does not mean a 95% chance the fixed interval contains the truth.
Means use the normal/t distribution (depending on n and σ known), while proportions use a normal approximation or exact methods when n·p or n·(1−p) are small.
Larger samples reduce the standard error, shrinking the interval width for the same confidence level.
One‑sample CIs estimate a single population mean/proportion; two‑sample CIs estimate a difference between two means/proportions.