# Confidence Interval Calculator

URL: https://speedycalculator.com/calculators/statistics/confidence-interval/
Last updated: 2026-10-25

> The confidence interval calculator estimates the range likely to contain the population mean, from the sample mean, standard deviation, sample size and confidence level. It uses the t distribution for small samples.

## Inputs

- Sample mean
- Standard deviation
- Sample size
- Confidence level

## Results

- Standard error
- Critical value (t*)
- Margin of error
- Lower bound
- Upper bound

## Formula

```
CI = x̄ ± t* × s ÷ √n
```

## Explanation

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

FieldWhat to enterSample meanA numberStandard deviationA numberSample sizeA numberConfidence levelChoose 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 meanLower bound36.2533.627554.3851.752572.569.877590.6388.0025108.75106.1275145142.3775

## Worked 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.

## FAQ

**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.


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)