How to Calculate Confidence Interval
What is Confidence Interval?
A confidence interval (CI) gives a range within which the true population parameter falls with a specified probability. A 95% CI means: if the experiment were repeated many times, 95% of CIs would contain the true value.
Formula
CI = x̄ ± z × (σ/√n), where z=1.96 for 95% CI, z=2.576 for 99% CI
- x̄
- Sample mean (value)
- σ
- Standard deviation (value)
- n
- Sample size (count)
- z
- Z-score (standard deviations)
Step-by-Step Guide
- 1CI = x̄ ± z × (σ/√n)
- 2z = 1.96 for 95% CI; 2.576 for 99% CI
- 3Width depends on sample size n and standard deviation σ
- 4Larger sample → narrower interval → more precise estimate
Worked Examples
Input
Mean 50, SD 10, n=100, 95% CI
Result
CI = 50 ± 1.96×(10/√100) = 50 ± 1.96 = [48.04, 51.96]
Frequently Asked Questions
What does a 95% confidence interval mean?
If repeated many times, 95% of intervals would contain the true population parameter. Not 95% chance this specific interval does.
When should I use 99% vs 95% CI?
99% is more conservative (wider). Use when high certainty needed (medical, safety). 95% standard in most research.
How does sample size affect confidence intervals?
Larger samples narrow the interval (more precision). √n in denominator means doubling n tightens by √2.