Confidence Interval Calculator
Compute the two-sided confidence interval for a population mean based on sample statistics. Estimates the plausibility range enclosing the true population mean at your chosen level of confidence.
The observed arithmetic mean of your sample
Sample or known population dispersion
Number of observations in the sample
Probability that interval covers true population parameter
Range expected to contain the true population mean
How to Use This Calculator
- Enter the observed sample mean.
- Enter the sample standard deviation.
- Enter the sample size (n).
- Select your confidence level (95% standard).
- Review the lower bound, upper bound, and margin of error.
Formula & Calculation Methodology
The margin of error equals the critical value Z multiplied by the standard error of the mean (SE = s / √n). As sample size n increases, the standard error shrinks.
Worked Example
Sample of 100 students: Mean 50, SD 10 at 95% confidence
Polling 400 voters: Mean 60, SD 20 at 99% confidence
Frequently Asked Questions
What does a 95% confidence interval actually mean?↓
It means that if we repeated the experiment 100 times under identical conditions, approximately 95 of the resulting confidence intervals would contain the true population mean.
Why does a higher confidence level widen the interval?↓
To be 99% certain of catching the true mean, you must cast a wider net than if you only require 90% certainty, increasing the critical Z value from 1.645 to 2.576.
How does increasing sample size affect the margin of error?↓
Margin of error is inversely proportional to the square root of n (√n). Quadrupling your sample size cuts the margin of error in half.
What is a limitation of using Z for small samples?↓
When sample size is under 30 and population variance is unknown, Student’s t-distribution provides slightly more conservative confidence bands.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.