A confidence interval gives a range around an estimate that captures the true population parameter with a specified probability — typically 95%. The Toolenza calculator returns the interval for a sample mean: CI = x̄ ± z × (σ / √n) for known population standard deviation, or the Student-t version for sample-based estimates.

How to read a 95% CI

The correct interpretation: "If we repeated this study many times and computed a 95% CI each time, 95% of those intervals would contain the true population value."

The wrong-but-common interpretation: "There's a 95% probability the true value is in this specific interval." This is the Bayesian credible-interval interpretation, which requires a prior — confidence intervals (frequentist) don't make that claim about a specific interval.

In practice, the two interpretations diverge less than the philosophical battle suggests; for most well-designed studies the numerical result is close.

Reading width

• Wide CI → low precision. You don't really know.
• Narrow CI → high precision. The estimate is well-measured.
• CI that includes zero → cannot rule out no effect.
• CI that doesn't include zero → equivalent to p < 1−level (so a 95% CI excluding zero implies p < 0.05).

Confidence levels you'll see

• 80% — z = 1.282. Often used in exploratory data analysis.
• 90% — z = 1.645. Common in epidemiology.
• 95% — z = 1.96. The default everywhere.
• 99% — z = 2.576. Required when stakes are high (drug approval, criminal forensics).

Pitfalls

The formula assumes the sample mean is roughly normally distributed — which requires either a normal population OR a large enough sample (n ≥ 30 by convention). For small samples from non-normal populations, use bootstrapping for a more honest interval.