Confidence Interval Calculator
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What Your Result Means
- Confidence Interval: A range around the sample mean that, at the chosen confidence level, is expected to contain the true population mean. A 95% CI means that if you repeated the sampling many times, about 95% of the intervals would capture the true mean.
- Margin of Error: The half-width of the interval — how far above and below the sample mean the interval extends. Smaller margins mean more precise estimates.
- Standard Error: The estimated standard deviation of the sample mean itself (not of individual data points). It shrinks as your sample size grows, because larger samples give more stable averages.
How This Calculator Works
You enter a sample mean, standard deviation, and sample size. The tool divides the standard deviation by the square root of the sample size to get the standard error, multiplies by the z-critical value for your confidence level, and adds/subtracts that margin from the mean. It uses z-scores (normal distribution), which is appropriate for large samples or known population standard deviation.
Quick Questions
When should I use a t-distribution instead?
Use a t-distribution when your sample size is small (typically under 30) and the population standard deviation is unknown. The t-distribution produces wider intervals to account for the extra uncertainty.
Does a 95% CI mean there's a 95% chance the true mean is inside it?
Not exactly. The true mean is a fixed value. A 95% CI means that 95% of intervals constructed this way from repeated samples would contain the true mean. Any single interval either does or doesn't contain it.
How can I make my confidence interval narrower?
Increase your sample size (which reduces the standard error), accept a lower confidence level, or reduce measurement variability. Quadrupling the sample size roughly halves the margin of error.
What if my data is not normally distributed?
For large samples (typically n > 30), the Central Limit Theorem says the sampling distribution of the mean is approximately normal regardless of the underlying distribution, so the z-based CI is still reasonable.
Can I use this for proportions (like survey percentages)?
This calculator is designed for means. For proportions, the standard error formula differs: SE = √(p(1−p)/n). Use a proportion-specific CI calculator for survey percentages.
Sources
- Wikipedia — Confidence Interval (definition, interpretation, and z vs. t approaches)
- Khan Academy — Confidence Intervals (step-by-step instructional resource)
- Stat Trek — Confidence Interval (formulas and z-score tables)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.