About this calculator
For a mean the calculator uses the t distribution when the standard deviation comes from the sample, which is the usual case.
How to use it
- Choose mean or proportion.
- Enter the sample results and confidence level.
- Read the interval and margin of error.
The formula
Mean: x̄ ± t × s ÷ √n; proportion: p̂ ± z × √(p̂(1 − p̂) ÷ n)
- x̄
- sample mean
- s
- standard deviation
- n
- sample size
- t, z
- critical value for the confidence level
Worked example
Mean 52.3, s 6.4, n 40 at 95%
- SE = 6.4 ÷ √40 = 1.012; t with 39 df = 2.023.
- Interval: 52.3 ± 2.047, or 50.25 to 54.35.
What the result means
95% confidence means that the method captures the true value in 95% of samples, not that there is a 95% chance this particular interval is right.
Assumptions
- A random sample; roughly normal data or a sample of 30 or more for a mean.
Limitations
- The Wald interval for proportions is poor for small samples; the Wilson interval is shown alongside.
Frequently asked questions
Should I use t or z?
Use t when the standard deviation is estimated from the sample. Use z only if the population standard deviation is genuinely known.
Why is the interval wider at 99%?
More confidence needs a wider net.
How do I make the interval narrower?
Increase the sample size: quadrupling n halves the width.
Last reviewed on 6 October 2026. Found a mistake? Tell us.