Confidence Interval Calculator
Free confidence interval calculator for means, proportions, and differences. Z-intervals, t-interval
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Confidence Interval โ Range of Plausible Values
Means, proportions, differences. Z-intervals, t-intervals. Correct interpretation: "We are 95% confident that..." โ the confidence is in the method.
Real-World Scenarios โ Click to Load
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways โ Correct CI Interpretation
- โข A confidence interval gives a range of plausible values for a parameter โ it does not mean "95% probability the true value is in the interval"
- โข Correct: "We are 95% confident that the true parameter lies in this interval" โ the confidence is in the method, not the specific interval
- โข Wrong: "There is 95% probability the true mean is in [a,b]" โ the parameter is fixed; the interval is random
- โข Use z-interval when population ฯ is known (rare) or for proportions; use t-interval when only sample SD is available (common)
- โข Margin of error = critical value ร standard error. Larger sample size โ smaller ME โ narrower CI
- โข CIs are NOT "95% of the data falls in this range" โ that would be a prediction interval
Did You Know?
How Confidence Intervals Work
A confidence interval is constructed from a point estimate (e.g., sample mean) plus or minus a margin of error. The margin of error depends on the standard error and the critical value (z* or t*).
Step 1: Choose Confidence Level
90%, 95%, or 99% are common. Higher confidence โ wider interval. 95% means: if we repeated the process many times, 95% of intervals would contain the true parameter.
Step 2: Get Critical Value (z* or t*)
z* from normal distribution (e.g., 1.96 for 95%). t* from t-distribution with df = nโ1 when ฯ is unknown. t* is larger than z* for small n, reflecting extra uncertainty.
Step 3: Compute Standard Error
SE = ฯ/โn (mean, ฯ known) or s/โn (mean, ฯ unknown) or โ(pฬ(1โpฬ)/n) (proportion).
Step 4: CI = Point Estimate ยฑ (Critical Value ร SE)
The margin of error is (z* or t*) ร SE. Add and subtract from the point estimate to get the lower and upper bounds.
Z vs t: When to Use Which
Use z (Normal)
When population ฯ is known (rare), or for proportions. Also for large n (n โฅ 30) as an approximation.
Use t (Student's t)
When you only have sample SD. Always correct for means with unknown ฯ, regardless of n. df = nโ1.
Frequently Asked Questions
What does '95% confident' mean?
It means that if we repeated the sampling process many times and constructed a 95% CI each time, about 95% of those intervals would contain the true parameter. It does NOT mean there is 95% probability that the current interval contains the true value.
When do I use z vs t?
Use z when the population standard deviation (ฯ) is known, or for proportions. Use t when you only have the sample standard deviation (s) for a mean. In practice, ฯ is rarely known, so t-intervals are more common for means.
Why is my t-interval wider than a z-interval?
The t-distribution has heavier tails than the normal distribution, especially for small df. So t* > z* for small samples, reflecting extra uncertainty when we estimate ฯ with s.
How do I calculate required sample size?
For a mean: n = (z* ร ฯ / ME)ยฒ. For a proportion: n = z*ยฒ ร p(1โp) / MEยฒ. Use p=0.5 if you have no prior estimate (maximizes required n).
Can a confidence interval include negative values?
Yes. For a difference (e.g., meanโ โ meanโ), if the CI includes 0, we cannot rule out that the true difference is zero. For proportions, the CI is truncated to [0, 1].
Official Data Sources
Disclaimer: Confidence intervals assume random sampling and approximate normality (for means) or large enough n for proportions (np โฅ 10, n(1โp) โฅ 10). For small or non-normal samples, results may be approximate. Consult a statistician for complex designs.
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