A 95% confidence interval means that if you repeated the study many times, about 95% of such intervals would contain the true mean.
How to use the Confidence Interval Calculator
- Sample mean
- Standard deviation
- Sample size
- Confidence level
Results update as you type, and every field accepts the units shown in its menu.
How it works
The critical value from the t distribution with n − 1 degrees of freedom is multiplied by the standard error.
Inputs explained
| Field | What to enter |
|---|---|
| Sample mean | A number |
| Standard deviation | A number |
| Sample size | A number |
| Confidence level | Choose one: 90%, 95%, 99% |
What each result means
- Standard error = [Standard deviation]÷sqrt([Sample size])
- Critical value (t*) = tinv(1 – (1 – [Confidence level])÷2, [Sample size] – 1)
- Margin of error = [Critical value (t×)]×[Standard error]
- Lower bound = [Sample mean] – [Margin of error]
- Upper bound = [Sample mean] + [Margin of error]
Example values
How the main result changes when Sample mean changes and everything else stays at the example values above:
| Sample mean | Lower bound |
|---|---|
| 36.25 | 33.6275 |
| 54.38 | 51.7525 |
| 72.5 | 69.8775 |
| 90.63 | 88.0025 |
| 108.75 | 106.1275 |
| 145 | 142.3775 |
Formula
CI = x̄ ± t* × s ÷ √nWorked example
Inputs — Sample mean: 72.5; Standard deviation: 8.2; Sample size: 40; Confidence level: 95%.
Results — Standard error = 1.2965; Critical value (t*) = 2.0227; Margin of error = 2.6225; Lower bound = 69.8775; Upper bound = 75.1225.
Frequently asked questions
Why use t instead of z?
t accounts for estimating the standard deviation from the sample; for large n they converge.
Does a wider interval mean worse data?
It means more uncertainty, from small samples or high variability.
What formula does the Confidence Interval Calculator use?
CI = x̄ ± t* × s ÷ √n.