Confidence Interval Calculator

This calculator computes confidence intervals for proportions (using the Wilson score method) or means (using the normal approximation). Confidence intervals provide a range of plausible values for the true population parameter.

Select the type of data
Data Type
Proportion (e.g., success rate, prevalence)
Mean (continuous data)
Number of Successes (Events) Count of positive outcomes
Sample Size (n) Total number of observations
Confidence Level

Formulas

Confidence Interval for a Proportion (Wilson Score)

\( \frac{\hat{p} + \frac{z^2}{2n} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n} + \frac{z^2}{4n^2}}}{1 + \frac{z^2}{n}} \)

Confidence Interval for a Mean

\( \bar{x} \pm z \cdot \frac{s}{\sqrt{n}} \)

where \( z \) is the critical value for the desired confidence level.

Understanding Confidence Intervals

A confidence interval provides a range of values that is likely to contain the true population parameter. A 95% confidence interval means that if we repeated the study many times, 95% of the calculated intervals would contain the true value.

The width of the confidence interval depends on:

  • Sample size — larger samples produce narrower intervals
  • Variability — more variable data produces wider intervals
  • Confidence level — higher confidence requires wider intervals

Wilson Score vs. Wald Interval

For proportions, this calculator uses the Wilson score interval rather than the simpler Wald interval. The Wilson method:

  • Provides better coverage, especially for small samples or extreme proportions
  • Never produces impossible values (below 0% or above 100%)
  • Is recommended by most statisticians for routine use

Assumptions & Limitations

  • For means: assumes data are approximately normally distributed (or n ≥ 30)
  • For proportions: Wilson method is robust but may be conservative for very small samples
  • Uses z-distribution; for small samples (n < 30), t-distribution may be more appropriate for means
  • Assumes simple random sampling

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