Confidence Interval Calculator
Last updated: November 5, 2026
Estimate a mean confidence interval from a sample mean, standard deviation, sample size and critical value.
Margin of error
—
- Lower confidence limit: —
- Upper confidence limit: —
- Standard error: —
- Interval width: —
On this page
Quick summary: Estimate a mean confidence interval from a sample mean, standard deviation, sample size and critical value.
Read the complete guide
Updated reference sections in this guide are in English.
What this calculator estimates
Estimate a mean confidence interval from a sample mean, standard deviation, sample size and critical value.
How to use it
Enter Sample mean; Sample standard deviation; Sample size; Critical value (z or t; 1.96 is a 95% z example). Select Calculate to update the results.
How the calculation works
The margin of error equals the critical value multiplied by the sample standard deviation divided by the square root of sample size.
Worked example: starting scenario
| Sample mean | 50 |
|---|---|
| Sample standard deviation | 12 |
| Sample size | 100 |
| Critical value (z or t; 1.96 is a 95% z example) | 1.96 |
| Margin of error | 2.352 |
|---|---|
| Lower confidence limit | 47.648 |
| Upper confidence limit | 52.352 |
| Standard error | 1.2 |
| Interval width | 4.704 |
Important limitations
This normal-approximation interval may be unsuitable for small or nonrepresentative samples.
Choose a critical value that matches the method
The arithmetic is mean ± critical value × standard deviation / √sample size. The form does not decide whether your sample supports a normal approximation or select a Student t value for you. For mean 50, standard deviation 10, sample size 100 and critical value 1.96, standard error is 1 and the interval is 48.04 to 51.96. A confidence level describes the method’s repeated-sampling coverage under its assumptions, not a guarantee that one sample is representative.