Build a simple analytics stack: Confidence Interval Calculator, Hypothesis Testing Calculator, Survey Question Randomizer, Quick Table Generator, BibTeX to Citation Converter, Word Counter, Confidence Interval Calculator Guide.
What this tool does
- One sample mean: Auto z or t based on sample context. You can also force z or t. Learn the difference in our Hypothesis Testing article.
- One sample proportion: Wilson and Agresti Coull for robust intervals, plus classic Wald for comparison.
- Two means: Welch for unequal variances or pooled for equal variances.
- Two proportions: Newcombe score difference or standard Wald difference.
- Bootstrap: Paste raw data and use percentile bootstrap for a distribution free view. Compare with simulations from the Random Data Generator.
- Sample size planning: Enter your target margin and confidence to compute n for a mean or proportion.
Why confidence intervals matter
Confidence intervals communicate both an estimate and its uncertainty. A narrow interval signals precision, while a wide interval highlights uncertainty. Unlike a single p value, an interval helps stakeholders judge practical significance. For example, a difference in means that excludes zero is informative, but the interval also shows if the effect is large enough to matter in a real setting. See APA statistical guidelines for best practices.
Method selection guide
- Mean with unknown sigma and moderate n: Use a t interval. The calculator does this automatically.
- Mean with known sigma or large n: A z interval is fine.
- Proportion with small or extreme counts: Prefer Wilson or Agresti Coull over Wald.
- Two means with unequal variances: Choose Welch. Use pooled only when equal variance is justified.
- Two proportions: Newcombe uses Wilson sub intervals to produce better coverage than simple Wald.
- Raw data with unknown shape: Add bootstrap to see a percentile interval alongside the parametric one.
Use 90 percent when speed and tighter margins are more valuable than coverage, such as quick product checks. Use 95 percent as a general default for reports and coursework. Use 99 percent when the cost of a wrong call is high. Remember that higher confidence increases the margin of error, so plan a larger sample if you need both high confidence and a tight margin.
Five step workflow
- Set the question: Mean or proportion, one or two groups. Define the metric and units. Use the Unit Converter to standardize measurements.
- Pick the level: Start at 95 percent unless your field specifies a different level.
- Choose the method: Follow the guide above. Prefer robust methods for small n or skewed counts.
- Compute and visualize: Run the calculator, review the interval, and view the mini chart to check width and center.
- Plan n if needed: If the margin is too wide, use the sample size planner to adjust n before collecting more data. For survey research, pair with the Survey Question Randomizer to balance responses.
Educational insights and pitfalls
- A 95 percent interval is about the process, not a single sample: If you repeated sampling many times, about 95 percent of the intervals would cover the true parameter.
- Do not treat the CI as a probability that the parameter lies inside: The parameter is fixed. The interval is random across repeated samples.
- Check independence: Intervals assume independent observations. Clustered or time series data need different methods. See our detailed CI guide.
- Beware small n for proportions: Wilson or Agresti Coull usually outperform Wald when counts are low or near 0 or 1.
- Variance equality is a claim, not a default: Use Welch unless you have good evidence that variances are equal.
- Report the interval and the method: Include estimate, lower bound, upper bound, level, and method. Example: mean 12.4, 95 percent CI [11.8, 13.0], Welch t.
Try it now
Open the Confidence Interval Calculator, set your confidence level, pick a method, and paste your data or summary stats. Then export a CSV for your report and cite the method used. For academic papers, pair it with the BibTeX Converter, format results in a table with the Quick Table Generator, and check readability with the Readability Score Checker.
FAQ
- Is the Confidence Interval Calculator free?
- Yes. The tool runs locally in your browser. Basic use is free and private.
- Does it choose z or t automatically for means?
- Yes. Auto mode uses t with sample standard deviation, or z when a known sigma or large n makes it appropriate. You can also force a method.
- Which method should I use for proportions?
- Wilson is a strong default. Agresti Coull is also robust. Use Wald mainly for comparison or when n is large and p is not extreme.
- Can I compare two groups?
- Yes. Use Welch or pooled for means and Newcombe or Wald for proportions. Welch and Newcombe are usually safer defaults.
- How do I plan sample size for a target margin?
- Use the planner inside the tool. Enter your target margin and confidence. Provide an estimated standard deviation for means or an estimated p for proportions.