What does "95% confident" actually mean?

95% confidence means that if you repeated the sampling process many times and built a CI each time, about 95% of those intervals would contain the true population parameter. It does NOT mean there is a 95% chance the parameter is in this specific interval.

How do I choose between 90%, 95%, and 99% confidence?

95% is the default in most fields. Use 99% when errors are costly (medical, safety). Use 90% for preliminary analysis or when narrower intervals are needed. Higher confidence = wider interval.

What makes a confidence interval narrower?

Three factors reduce CI width: (1) larger sample size n (most practical), (2) lower confidence level (trade-off with certainty), (3) smaller population variance σ² (often not controllable).

What sample size do I need for a ±3% margin of error?

For 95% confidence with p̂ ≈ 0.5 (worst case): n = (1.96/0.03)² × 0.25 ≈ 1,068. For more extreme proportions (e.g., p̂ ≈ 0.1), you need fewer: n ≈ 385.

When is the normal approximation valid for proportions?

When both np̂ ≥ 10 and n(1−p̂) ≥ 10. For smaller samples, use the Wilson score interval or exact binomial CI (Clopper-Pearson).

All Tools

Confidence Interval Calculator

Calculate confidence intervals for means and proportions with step-by-step solutions and Python code.

Enter Data

Sample Mean (x̄)

Sample Std Dev (s)

Sample Size (n)

Confidence Level

Sample Data:

Enter your data and click Calculate CI.

Common Critical Values:

90% CI: z* = 1.645
95% CI: z* = 1.960
99% CI: z* = 2.576

Frequently Asked Questions

When should I use t vs z for confidence intervals?

Use t when σ is unknown (you estimated it from the sample as s). Use z only when σ is known. In practice, always use t - it converges to z for large n, so it is never wrong to use t.

How do degrees of freedom affect the interval?

Lower df = wider interval. The t-distribution has heavier tails with fewer df, increasing the critical value. At df=10, t* ≈ 2.228 vs z* = 1.960 for 95% CI. At df=100, t* ≈ 1.984.

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