Confidence Interval Calculator - Z-Score Method
Calculate a confidence interval from sample mean, standard deviation, sample size, and confidence level.
95% Confidence Interval
[46.080, 53.920]
Margin of error: ±3.9200How to Calculate a Confidence Interval
A confidence interval gives a range around a sample mean that's likely to contain the true population mean. Enter your sample mean, sample standard deviation, sample size, and a confidence level, and this calculator finds the interval using the standard z-score approach.
Example
A sample mean of 50, a standard deviation of 10, and a sample size of 25 give a standard error of 10 ÷ √25 = 2. At 95% confidence (z = 1.96), the margin of error is 1.96 × 2 = 3.92, giving a confidence interval of roughly [46.08, 53.92].
Common Use Cases
- Reporting survey results with an appropriate margin of error.
- Summarizing A/B test or experiment results in statistics coursework or research.
- Estimating a plausible range for a population parameter from sample data.
FAQs
Why use a z-score instead of a t-score?
The z-score approximation is simpler and reasonably accurate for larger sample sizes (typically n ≥ 30). For smaller samples, a t-distribution (which has heavier tails to account for the extra uncertainty of estimating from limited data) is technically more precise — treat this z-score result as a solid approximation rather than an exact figure for small samples.
What does "95% confidence" actually mean?
It means that if you repeated the same sampling process many times and built a confidence interval each time, about 95% of those intervals would contain the true population mean. It's not a 95% probability that the true mean falls within this one specific interval you calculated.
How does sample size affect the interval?
Larger sample sizes shrink the standard error (since it's divided by the square root of n), which narrows the confidence interval and gives a more precise estimate of the population mean.