Confidence Interval Calculator
Enter your sample mean, standard deviation and sample size to get a confidence interval for the population mean at the 90%, 95% or 99% level.
Quick answer: A confidence interval for a mean is x̄ ± z·(σ ÷ √n). For x̄ = 100, σ = 15, n = 25 at 95% (z = 1.96): margin = 1.96 × (15 ÷ 5) = 5.88, so the interval is 94.12 to 105.88.
A confidence interval gives a plausible range for the true population mean based on your sample. A 95% interval means that if you repeated the sampling many times, about 95% of the intervals built this way would contain the real mean. The width comes from the standard error scaled by a critical value for the confidence level.
The formula
CI = x̄ ± z · (σ ÷ √n)
z: 90% = 1.645 · 95% = 1.96 · 99% = 2.576
z: 90% = 1.645 · 95% = 1.96 · 99% = 2.576
Worked example
x̄ = 100, σ = 15, n = 25, 95% level. Standard error = 15 ÷ √25 = 3. Margin = 1.96 × 3 = 5.88. Interval = 100 ± 5.88 = 94.12 to 105.88.
Frequently asked questions
What does a 95% confidence interval mean?
If you repeated the study many times, about 95% of the intervals computed this way would contain the true population mean.
Which z-value goes with each level?
90% uses 1.645, 95% uses 1.96, and 99% uses 2.576 for a two-sided interval.
When should I use a t-interval instead?
Use the t-distribution when the sample is small and the population standard deviation is unknown; it makes the interval a little wider.
How do I make the interval narrower?
Increase the sample size or accept a lower confidence level — both reduce the margin of error.
Related calculators
For general and educational use. Uses the normal (z) interval.