Confidence Interval Calculator
Enter your sample mean, standard deviation, sample size, and confidence level to calculate the confidence interval — the range likely to contain the true population mean.
How the Confidence Interval formula works
The confidence interval formula:
Margin of error = Z × (Standard deviation / √n) Confidence interval = Mean ± Margin of error
Z is the critical value for your confidence level (1.96 for 95%, 2.576 for 99%, 1.645 for 90%).
Step-by-step calculation
- Divide the sample standard deviation by the square root of the sample size to get the standard error.
- Multiply the standard error by the Z-score for your chosen confidence level to get the margin of error.
- Add and subtract the margin of error from the sample mean to get the interval's upper and lower bounds.
Worked example
A sample mean of 68, standard deviation of 12, sample size of 50, at 95% confidence: Standard error = 12/√50 ≈ 1.697. Margin of error = 1.96 × 1.697 ≈ 3.33. Confidence interval = 68 ± 3.33, or roughly 64.67 to 71.33.
Frequently asked questions
What does a 95% confidence interval actually mean?
It means that if you repeated the sampling process many times, about 95% of the resulting intervals would contain the true population mean — it's a statement about the reliability of the method, not a 95% probability that this specific interval contains the true value.
Why does a larger sample size narrow the interval?
Standard error decreases as sample size increases (it's divided by the square root of n), so larger samples produce more precise estimates and correspondingly narrower confidence intervals for the same confidence level.