How to Use the Confidence Interval Calculator

Choose the type of confidence interval you need: mean (for continuous data), proportion (for success/failure data), difference (comparing two groups), or sample size (planning studies).

Confidence Interval Types

Mean Confidence Interval

Estimates the range where the true population mean likely falls:

Formula: x̄ ± t × (s/√n)
Uses t-distribution for small samples (n < 30)
Uses z-distribution for large samples or known σ

Proportion Confidence Interval

Estimates the range for a population proportion using the Wilson score interval:

More accurate than the Wald interval for extreme proportions
Works well even with small sample sizes
Never produces intervals outside [0, 1]

Difference of Means

Uses Welch's t-interval for comparing two independent groups:

Doesn't assume equal variances
More robust than pooled t-test
If CI includes 0, difference is not significant

Sample Size Estimation

Calculate how many observations you need for a desired precision:

Smaller margin of error → larger sample needed
Higher confidence level → larger sample needed
Requires estimated standard deviation

Interpreting Confidence Intervals

95% CI: If we repeated the study many times, 95% of CIs would contain the true value
Width: Narrower intervals indicate more precise estimates
Significance: If a CI for difference doesn't include 0, the difference is significant

Common Confidence Levels

90%: z = 1.645 (less strict)
95%: z = 1.96 (most common)
99%: z = 2.576 (more conservative)
99.9%: z = 3.291 (very conservative)

Important Considerations

Sample size matters: Larger samples produce narrower intervals
Assumptions: Most CIs assume random sampling
Not probability: The CI either contains the true value or it doesn't
Margin of error: Half the width of the confidence interval

Applications

Survey research and polling
Clinical trials and medical studies
Quality control and manufacturing
A/B testing and marketing experiments
Scientific research and hypothesis testing