Confidence Interval Calculator
Population Mean Estimation Tool
Understanding Confidence Intervals
A confidence interval turns a single sample estimate into a range, giving you a sense of how precisely your sample data pins down the true population mean.
The margin of error combines your sample's spread (standard deviation) with your sample size and your chosen confidence level into a single ± value added to and subtracted from the sample mean.
The confidence level describes the long-run reliability of the method, not the probability for one specific interval. Higher confidence (99% vs. 90%) requires a wider interval to maintain that reliability.
The Formula
Where x̄ is the sample mean, z is the critical value for your confidence level, σ is the sample standard deviation, and n is the sample size.
A Note on z vs. t
This calculator uses the standard z-critical values (1.645, 1.96, 2.576), which treat the calculation as a large-sample or known-population-variance approximation — the standard, widely-used approach for a general-purpose confidence interval tool. For rigorous work with small samples and an unknown population standard deviation, statisticians typically switch to a t-distribution with n − 1 degrees of freedom, which has slightly heavier tails to account for the added uncertainty. As sample size grows past roughly 30, the difference between the z and t results becomes negligible.
Narrowest interval of the three presets.
The most commonly used confidence level.
Widest interval — highest reliability of the three presets.
Key Takeaways
- A confidence interval describes the reliability of the method, not the probability that one specific interval contains the true mean.
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Larger samples produce tighter intervals — the margin of error shrinks as
ngrows, since it's divided by√n. - Higher confidence means a wider interval. There's an inherent trade-off between confidence and precision.
- Need the underlying z-score or percentile math? The Z-score Calculator covers the standard normal distribution in more depth.
Frequently Asked Questions
- Enter your sample mean.
- Enter your sample standard deviation.
- Enter your sample size (n).
- Choose a confidence level — 90%, 95%, or 99%.
- Click "Calculate" to see the margin of error and the full confidence interval.
A confidence interval is a range of values, built from sample data, that's likely to contain the true population mean. A 95% confidence interval means that if you repeated the same sampling process many times and built an interval each time, about 95% of those intervals would contain the true population mean — it does not mean there's a 95% chance the true mean falls in this one specific interval.
They're the standard critical values from the normal distribution for the most common confidence levels: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. Each value marks how many standard errors you need to travel from the mean so that the specified percentage of a normal distribution falls between the two bounds.
This calculator uses the large-sample (or known-population-variance) approximation, which is standard practice for a general-purpose web calculator. For small samples (commonly n < 30) with an unknown population standard deviation, a t-distribution with n − 1 degrees of freedom is technically more accurate, since it accounts for the extra uncertainty of estimating variance from a small sample. As the sample size grows, the t-distribution converges to the normal distribution, so the z-based result shown here becomes an increasingly close approximation.
The margin of error is the ± amount added to and subtracted from the sample mean to build the interval: margin of error = z × (σ ÷ √n). A larger sample size (n) or a smaller standard deviation shrinks the margin of error, producing a tighter, more precise interval.
To be more confident that the interval captures the true population mean, you need to cast a wider net. That's why the z-value — and therefore the margin of error and the width of the interval — increases as you move from 90% to 95% to 99% confidence.