Confidence Interval Calculator

A confidence interval is a range of values that likely contains the true population parameter. A 95% CI means: if you repeated the study many times, 95% of the calculated intervals would contain the true mean.

Use the t-distribution when the sample size is small (n < 30) or when the population standard deviation is unknown (which is almost always). For n >= 30, the z and t distributions are nearly identical.

The margin of error is the half-width of the confidence interval: ME = critical value × standard error. A smaller sample size or lower confidence level gives a smaller margin of error, but less precision.

Standard error (SE = σ/sqrt(n)) measures how much the sample mean varies from sample to sample. As sample size increases, the standard error decreases — larger samples produce more precise estimates.

A wider interval means less precision, not less accuracy. It means there is more uncertainty about where the true parameter lies. You can narrow the interval by increasing sample size or lowering the confidence level.