Confidence Interval Calculator

A confidence interval is a range likely to contain the true mean.

How the Math Works

A confidence interval for a mean estimates the range in which the true population average likely falls, using the formula CI = mean ± z × σ ÷ √n. Here, the sample mean (μ) represents the center of the interval, while the margin of error (z σ/√n) accounts for variability: z is the critical value from the standard normal distribution (e.g., 1.96 for 95% confidence), σ is the population standard deviation, and n is the sample size. The larger your sample (n), the smaller the standard error (σ/√n), narrowing the interval and increasing precision. This formula assumes random sampling and a roughly normal population or large enough sample size for the Central Limit Theorem to apply.

Practical Applications

To apply this calculation, first gather your sample data: compute the sample mean, determine the known population standard deviation (or use sample standard deviation with a t-distribution for small samples), and note your sample size. Choose your desired confidence level (common choices are 90%, 95%, or 99%), then find the corresponding z-score. Plug these values into the formula to calculate the lower and upper bounds of your interval. For example, if testing a new drug's effectiveness with a sample mean reduction of 5 points, σ = 10, n = 100, and 95% confidence (z = 1.96), the interval would be 5 ± 1.96, giving you a range of 1.96 to 8.04 points where the true effect likely lies.

Day-to-Day Use

Confidence intervals help you make informed decisions beyond just point estimates in everyday situations. When a political poll shows a candidate leading by 4 points with a 95% confidence interval of 4 ± 3 points (-1 to 9), you realize the race is actually too close to call. Quality control managers use them to determine if a production batch meets specifications, while doctors interpret treatment effectiveness ranges rather than relying on single averages. Even in personal finance, understanding the confidence interval around expected investment returns helps you plan more realistically—knowing returns likely fall between 6% and 10% instead of just expecting exactly 8% guides better financial planning and risk management.

Worked example

mean 100, σ 15, n 36, 95% → 95.1 to 104.9.

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