Confidence Interval Calculator
Calculate a confidence interval for a sample mean, given the sample's standard deviation and size.
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- Works on all devices
Enter your sample mean, standard deviation, size, and confidence level to find the confidence interval.
Your result
4.9
Margin of error
Lower bound95.1
Upper bound104.9
AI explanation
Formula
Margin of error = Z × (standard deviation ÷ √sample size); interval = mean ± margin of errorWorked example
Mean 100, std dev 15, n=36, 95% confidence
| Field | Value |
|---|---|
| Sample mean | 100 |
| Sample standard deviation | 15 |
| Sample size | 36 |
| Confidence level | 95 |
| Margin of error | 4.9 |
| Lower bound | 95.1 |
| Upper bound | 104.9 |
Assumptions
- Uses the normal (Z) distribution approximation rather than the t-distribution — a reasonable approximation once the sample size is reasonably large (a common rule of thumb is n ≥ 30). For small samples, a t-distribution-based interval would be more precise.
Frequently asked questions
What does a 95% confidence interval mean?
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 about this specific interval.
Why does a larger sample size narrow the interval?
The margin of error shrinks with the square root of the sample size — more data gives a more precise estimate of the true mean.
Should I use the Z or t distribution?
The Z distribution (used here) is a good approximation for sample sizes of about 30 or more; smaller samples technically need the t-distribution, which has fatter tails to account for the extra uncertainty.