Confidence Interval Calculator (Mean)
By Santaji GadepatilinUpdated October 27, 2026
A confidence interval gives a range that is likely to contain the true population mean, based on a sample. It shows how precise an estimate is.
How to use the confidence interval calculator
Choose a confidence level, then enter the sample mean, standard deviation and sample size. The margin of error and interval bounds are calculated.
Confidence interval formula
CI = x̄ ± z × s ÷ √n
Worked example
Mean 50, SD 12, n = 100, 95% confidence:
- Margin of error = 1.96 × 12 ÷ 10 = 2.35
- Interval = 47.65 to 52.35
Confidence levels and z-values
| Confidence | z* |
|---|---|
| 80% | 1.282 |
| 90% | 1.645 |
| 95% | 1.960 |
| 98% | 2.326 |
| 99% | 2.576 |
| 99.9% | 3.291 |
Small samples: use t instead of z
For small samples (n below about 30) with an unknown population standard deviation, use the t-distribution with n − 1 degrees of freedom:
| n | t* for 95% |
|---|---|
| 5 | 2.776 |
| 10 | 2.262 |
| 15 | 2.145 |
| 20 | 2.093 |
| 30 | 2.045 |
| 60 | 2.001 |
| Large | 1.960 |
Replace z* with t* in the formula to get a wider, more honest interval.
What a 95% interval means
If you repeated the study many times, about 95% of the intervals built this way would contain the true mean. It does not mean there is a 95% probability that the true mean lies in this particular interval.
Narrowing the interval
- Larger sample: the margin shrinks with √n — four times the sample halves the margin.
- Lower confidence: 90% gives a narrower interval than 99%, at the cost of more risk.
- Less variable data: better measurement reduces the standard deviation.
Proportions
For percentages (such as survey results) use the margin of error calculator, which uses the standard error of a proportion.
z values
| Confidence | z |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 98% | 2.326 |
| 99% | 2.576 |
Frequently asked questions
What does a 95% confidence interval mean?
If you repeated the sampling many times, about 95% of the intervals built this way would contain the true population mean.
How do I make the interval narrower?
Increase the sample size (quadrupling n halves the width) or accept a lower confidence level.
Should I use z or t?
This uses z, which is accurate for samples of about 30 or more. For small samples, a t-interval is wider.
What is a 95% confidence interval?
A range, built from sample data, that would contain the true population value in about 95% of repeated samples.
Should I use z or t?
Use t when the sample is small and the population standard deviation is estimated from the sample. For large samples they give almost the same result.