Confidence Interval Calculator
Calculate confidence intervals for means and proportions.
What Is the Confidence Interval Calculator?
The Confidence Interval Calculator computes a range of values likely to contain the true population mean. You provide the sample mean, standard deviation, sample size, and desired confidence level. The tool returns the margin of error and the lower and upper bounds of the interval.
Formula
How to Use
Enter the sample mean (x̄), the standard deviation (σ or s), the sample size n, and select the confidence level. Click Calculate to see the margin of error and the confidence interval [lower, upper].
Example Calculation
Survey of n=50 people, mean response x̄=74, std dev σ=12 95% confidence: Z = 1.960 SE = 12/√50 = 12/7.071 = 1.697 Margin of Error = 1.960 × 1.697 = 3.327 CI = [74 − 3.327, 74 + 3.327] = [70.67, 77.33] Interpretation: We are 95% confident the true mean lies between 70.67 and 77.33.
Understanding Confidence Interval
Confidence intervals are the cornerstone of inferential statistics. They quantify the precision of an estimate and communicate uncertainty — far more informative than a single point estimate alone.
The width of a confidence interval depends on three factors: variability (σ), sample size (n), and confidence level. Doubling the sample size reduces the interval width by a factor of √2 ≈ 1.41. To halve the margin of error, you need to quadruple the sample size.
Confidence intervals appear in almost every published scientific study. Drug trials report 95% CIs for treatment effects. Economic reports include CIs for GDP growth estimates. Clinical guidelines use CIs to determine whether a treatment effect is practically significant.
Frequently Asked Questions
What does a 95% confidence interval mean?
If you repeated the sampling process many times, 95% of the resulting confidence intervals would contain the true population mean. It does NOT mean there is a 95% chance the true mean is in this specific interval.
What is the standard error?
The standard error (SE = σ/√n) measures the variability of the sample mean across repeated samples. A larger n reduces the standard error, making the interval narrower and more precise.
Should I use z or t distribution?
Use z (this calculator) when n ≥ 30 or σ is known. Use t when n < 30 and σ is estimated from the sample — the t-distribution has heavier tails to account for greater uncertainty.
How do I reduce the margin of error?
Increase the sample size n (most effective), decrease the confidence level (less certainty), or reduce variability (better measurement methods or more homogeneous population).
What is the relationship between CI and hypothesis testing?
A 95% CI is the set of all null hypothesis values that would not be rejected at the α = 0.05 level. If the hypothesized value falls outside the CI, reject the null hypothesis.