Confidence Interval Calculator
Calculate confidence intervals for population means with z-scores.
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Quick Tip
Enter the sample mean (x̄).
About This Tool
Why use Confidence Interval?
A confidence interval gives a range that is likely to contain the true population mean: x̄ ± z × (σ ÷ √n). Enter your sample mean, standard deviation and sample size, choose a confidence level, and the calculator returns the interval, the margin of error, the standard error and the z-score it used.
Example: a mean of 50 with σ = 10 and n = 100 has a standard error of 1. At 95% confidence the z-score is 1.96, so the margin of error is 1.96 and the interval is 48.04 to 51.96. Because the margin depends on √n, cutting it in half takes four times as many observations.
Key Benefits
Standard error shown separately
Seeing σ ÷ √n on its own makes it clear how much of the margin comes from sample size.
Common confidence levels built in
The matching z-score (1.645, 1.96, 2.576) is applied and displayed, so there is no table to consult.
Shows the cost of more certainty
Switching from 95% to 99% widens the interval immediately, which is the trade-off the level controls.
How to use Confidence Interval
- 1
Enter the sample mean (x̄).
- 2
Enter the standard deviation (σ) and the sample size (n).
- 3
Pick a confidence level — 95% is the usual default; 99% gives a wider interval.
- 4
Read the interval, and use the margin of error when reporting the result as "50 ± 1.96".
Best use cases
Report survey results with a margin of error.
Estimate a population average from a sample measurement.
Decide how large a sample needs to be for an acceptable margin.
Check confidence interval questions in statistics coursework.
Confidence Interval FAQs
What does a 95% confidence interval actually mean?
If you repeated the same sampling process many times, about 95% of the intervals produced would contain the true population mean. It is not a 95% probability that the mean lies in this particular interval.
Which z-score goes with which confidence level?
1.645 for 90%, 1.96 for 95% and 2.576 for 99%. Higher confidence means a larger z and a wider interval.
Should I use z or t?
This calculator uses z, which is appropriate for large samples (roughly n ≥ 30) or when the population standard deviation is known. For small samples with an estimated standard deviation, the t-distribution gives a slightly wider, more accurate interval.
How do I shrink the margin of error?
Increase the sample size or accept a lower confidence level. Because the margin scales with 1 ÷ √n, halving it requires four times the sample.
Related Statistics Tools
More free statistics tools that work well alongside confidence interval.