Calculator Form
Plotly Graph
Submit the form to generate a method comparison confidence interval chart.
Example Data Table
| Trials | Successes | Point Estimate | Confidence Level | Method | Lower Bound | Upper Bound |
|---|---|---|---|---|---|---|
| 40 | 18 | 45.00% | 90% | Wald | 32.06% | 57.94% |
| 50 | 32 | 64.00% | 95% | Wilson | 50.14% | 75.86% |
| 80 | 45 | 56.25% | 90% | Agresti-Coull | 47.07% | 65.02% |
| 120 | 72 | 60.00% | 99% | Wilson | 48.25% | 70.70% |
Formula Used
Point estimate: p̂ = x / n, where x is successes and n is trials.
Wald interval: p̂ ± z × √[p̂(1 - p̂) / n]
Wilson interval: Uses an adjusted center and adjusted margin. It usually performs better when samples are small or probabilities are extreme.
Agresti-Coull interval: Adds z² pseudo-observations, then computes p̃ ± z × √[p̃(1 - p̃) / ñ]
Margin of error: The half-width of the interval. Larger samples reduce this width.
Interpretation: A 95% confidence interval means the method would capture the true probability in about 95% of repeated samples.
How to Use This Calculator
- Enter the number of successes observed.
- Enter the total number of trials collected.
- Choose your desired confidence level.
- Select the interval method to apply.
- Choose how many decimals to display.
- Press Calculate Interval to view results.
- Review the interval, method comparison, and chart.
- Use the export buttons to save the report.
FAQs
1. What does this calculator estimate?
It estimates a confidence interval for a population probability or proportion. You enter successes and total trials, then the tool returns a plausible range for the true underlying probability.
2. What is a point estimate?
The point estimate is the observed sample proportion. It equals successes divided by trials. It gives the best single-number estimate before uncertainty is added through the interval.
3. Why are there multiple interval methods?
Different methods behave differently under small samples or extreme proportions. Wald is simple. Wilson and Agresti-Coull often provide more stable intervals when the sample is limited or the estimate is near zero or one.
4. Which method should I choose?
Wilson is often a strong default. It usually performs better than Wald for many practical datasets. Agresti-Coull is also robust and easy to interpret.
5. What does confidence level mean?
The confidence level describes long-run method performance. A 95% interval means that if you repeated the study many times, about 95% of intervals built the same way would contain the true probability.
6. Why does the interval get narrower with larger samples?
Larger samples reduce sampling variability. That lowers the standard error and the margin of error, so the resulting interval becomes tighter and more informative.
7. Can I use this for percentages?
Yes. The calculator computes using proportions internally, then displays results as percentages. A probability of 0.64 is shown as 64%.
8. When should I avoid the Wald method?
Avoid Wald when the sample is small or the proportion is very close to zero or one. In those cases, Wilson or Agresti-Coull usually gives more reliable coverage.