95% Confidence Interval Calculator
Calculate a 95% confidence interval for a population mean from your sample statistics. Automatically selects the z-score (for large samples) or t-score (for small samples) method.
A 95% confidence interval is the most commonly used interval in statistical analysis. It gives a range where you can be 95% confident the true population mean falls.
The Formula:
CI = x̄ +/- critical value x (s / sqrt(n))
For 95% confidence:
- z* = 1.96 (when population SD is known or n >= 30)
- t* varies by degrees of freedom (when population SD is unknown and n < 30)
When to Use Z vs. T:
| Situation | Method | Critical Value |
|---|---|---|
| Known sigma, any n | Z | 1.96 |
| Unknown sigma, n >= 30 | Z (approximate) | 1.96 |
| Unknown sigma, n < 30 | T | Depends on df |
How to Make the Interval Narrower:
- Increase sample size (most effective)
- Reduce variability in measurements
- Accept a lower confidence level (e.g., 90%)
Example: A sample of 40 test scores has mean 72.5 and SD 8.3. The 95% CI is 72.5 +/- 1.96 x (8.3 / sqrt(40)) = 72.5 +/- 2.57, giving the interval [69.93, 75.07].