Math
Confidence Interval Calculator
Build a confidence interval for a mean, proportion, or difference of two means. Picks z or t, supports 80, 90, 95, 98, 99, and 99.9 percent.
Pick a calculation
Uses t when n is below 30; z otherwise.
95% confidence interval
[95.8422, 104.1578]
We are 95% confident that the true population mean lies between 95.8422 and 104.1578.
Sample mean
100
Margin of error
4.1578
Standard error
2.1213
Critical z*
1.96
Frequently Asked Questions about the Confidence Interval Calculator
What does a 95% confidence interval actually mean?
It is a range built from your sample using a procedure that, if repeated on many independent samples, would capture the true population value about 95% of the time. It is not the probability that the true value is inside this one specific interval; the 95% describes the long-run reliability of the method.
When does the calculator use z and when does it use t?
For a single-sample mean it uses Student's t when n < 30 (df = n - 1) and switches to z for n >= 30. For a proportion or a difference of two means it uses z (normal approximation). Treat small-sample proportion and difference results as approximate.
What is the margin of error?
Margin of error is critical value times standard error, the half-width of the interval. A 95% CI of 100 plus or minus 4 has a margin of error of 4. Shrinking it requires a larger sample, a lower confidence level, or less variability.
How do I pick a confidence level?
95% is the convention in most research. Use 90% when a narrower interval matters more than rigor. Use 99% or 99.9% when the cost of missing the true value is high, such as clinical or safety contexts. Higher confidence always means a wider interval for the same sample.
Why warn about small samples for proportions?
The Wald interval relies on a normal approximation to the binomial. It performs poorly when np or n(1-p) is below about 10. In that regime use Wilson score or Clopper-Pearson exact intervals instead.