About the Confidence Interval Calculator
A confidence interval gives a range of plausible values for a population parameter based on your sample, rather than a single point estimate. It is the honest way to report a result, because it makes the uncertainty in the estimate explicit.
The formula
CI = x̄ ± z · (σ / √n)x̄ is the sample mean, σ the standard deviation, n the sample size, and z the critical value for your confidence level — 1.96 for 95 percent. The term after ± is the margin of error.
How to use this calculator
- 1Enter your Sample Mean (x̄). The field starts at
50, which you can overwrite. - 2Enter your Standard Deviation (σ). The field starts at
10, which you can overwrite. - 3Enter your Sample Size (n). The field starts at
30, which you can overwrite. - 4Enter your Confidence Level (90, 95, or 99). The field starts at
95, which you can overwrite. - 5Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Sample Mean (x̄) | 50 |
| Standard Deviation (σ) | 10 |
| Sample Size (n) | 30 |
| Confidence Level (90, 95, or 99) | 95 |
Result
95% CI: [46.4215, 53.5785]
Margin of Error: ±3.5785
Standard Error: 1.8257
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
The interval narrows with the square root of sample size, which sets the economics of research. Quadrupling your sample halves the margin of error, so precision gets steadily more expensive. Going from 100 to 400 respondents halves the width; halving it again needs 1,600.
The interpretation is more subtle than it looks. A 95 percent confidence interval does not mean there is a 95 percent probability the parameter lies in this particular range. It means that if you repeated the study many times, 95 percent of the intervals produced would contain the true value. The confidence is in the procedure, not in any single interval.
Things worth knowing
- Wider confidence means a wider interval. A 99 percent interval is necessarily less precise than a 95 percent one.
- For small samples with unknown population standard deviation, the t-distribution gives more accurate intervals than z.
- If two intervals overlap substantially, the difference between the groups is probably not statistically significant.
- An interval that includes zero for a difference means you cannot rule out no effect at all.
- Always report the interval, not just the point estimate. A mean without uncertainty is an incomplete result.
Frequently asked questions
What does a 95 percent confidence interval mean?+
That the method producing it captures the true population value 95 percent of the time across repeated samples. It is a statement about the reliability of the procedure, not a probability about this specific interval.
How do I make a confidence interval narrower?+
Increase the sample size, which shrinks the margin by the square root of n, or accept a lower confidence level. Reducing variability in measurement also helps where that is possible.
When should I use t instead of z?+
When the population standard deviation is unknown and the sample is small, roughly under 30. The t-distribution has heavier tails, producing wider and more honest intervals for small samples.
What does it mean if two intervals overlap?+
That the difference between the groups may not be statistically significant. Substantial overlap suggests the data cannot distinguish them, though a formal comparison test is the proper check.