Confidence Interval Calculator
Confidence Interval Calculator
Enter your sample's mean, standard deviation, and size, plus your desired confidence level. This calculates the range of values that likely contains the true population mean, using the Student's t distribution (the correct choice when the population standard deviation is unknown and estimated from the sample, as is almost always the case in practice).
- Formula: CI = sample mean ± (t-critical × standard deviation ÷ √n), where the t-critical value depends on both your confidence level and the sample's degrees of freedom (n − 1).
- Wider isn't worse: a wider interval at the same confidence level usually just reflects more variability or a smaller sample — it's not a sign of a flawed calculation.
- Uses the t distribution, not a fixed z-value: smaller samples get a proportionally wider interval to account for the extra uncertainty in estimating the standard deviation itself — the t-critical value converges toward the familiar z-values (1.96 for 95%) as sample size grows.
What does "95% confidence" actually mean?
It means that if you repeated this sampling process many times and built a confidence interval each time, about 95% of those intervals would contain the true population mean — it's a statement about the method's long-run reliability, not the probability that this specific interval is correct.
Why does a smaller sample size widen the interval?
Smaller samples give a less precise estimate of both the mean and the standard deviation, so the t distribution (used here) assigns a larger critical value to compensate for that extra uncertainty, compared to the normal distribution.
Confidence Interval Calculator
Enter your sample's mean, standard deviation, and size, plus your desired confidence level. This calculates the range of values that likely contains the true population mean, using the Student's t distribution (the correct choice when the population standard deviation is unknown and estimated from the sample, as is almost always the case in practice).
- Formula: CI = sample mean ± (t-critical × standard deviation ÷ √n), where the t-critical value depends on both your confidence level and the sample's degrees of freedom (n − 1).
- Wider isn't worse: a wider interval at the same confidence level usually just reflects more variability or a smaller sample — it's not a sign of a flawed calculation.
- Uses the t distribution, not a fixed z-value: smaller samples get a proportionally wider interval to account for the extra uncertainty in estimating the standard deviation itself — the t-critical value converges toward the familiar z-values (1.96 for 95%) as sample size grows.
What does "95% confidence" actually mean?
It means that if you repeated this sampling process many times and built a confidence interval each time, about 95% of those intervals would contain the true population mean — it's a statement about the method's long-run reliability, not the probability that this specific interval is correct.
Why does a smaller sample size widen the interval?
Smaller samples give a less precise estimate of both the mean and the standard deviation, so the t distribution (used here) assigns a larger critical value to compensate for that extra uncertainty, compared to the normal distribution.