Confidence Interval Calculator

Calculate confidence intervals for population means and proportions based on sample data.

Calculation Settings

Confidence Interval Formulas

For Proportion

The formula for a confidence interval for a proportion is:

p^±zα/2p^(1−p^)n\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

Where:

  • p^\hat{p} = Sample proportion
  • zÎą/2z_{\alpha/2} = Critical value from the standard normal distribution
  • nn = Sample size

For Mean (Normal Distribution)

When the population standard deviation is known or the sample size is large:

xˉ±zÎą/2Īƒn\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}

Where:

  • xˉ\bar{x} = Sample mean
  • zÎą/2z_{\alpha/2} = Critical value from the standard normal distribution
  • Īƒ\sigma = Population standard deviation
  • nn = Sample size

For Mean (t-Distribution)

When the population standard deviation is unknown and the sample size is small:

xˉ±tα/2,n−1sn\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}

Where:

  • xˉ\bar{x} = Sample mean
  • tÎą/2,n−1t_{\alpha/2, n-1} = Critical value from the t-distribution with n-1 degrees of freedom
  • ss = Sample standard deviation
  • nn = Sample size

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