Quickly compute two-sided confidence intervals for sample means and proportions. Enter your sample size, summary statistics, and confidence level to get an accurate range that likely contains the true population parameter.
What is a Confidence Interval?
A confidence interval (CI) is a range of values that estimates an unknown population parameter, such as a mean or proportion. Instead of giving a single best guess, a CI provides a lower and upper bound that reflects sampling uncertainty. A 95% CI means that if you repeated the same study many times, approximately 95% of the calculated intervals would contain the true parameter.
When to Use the Confidence Interval Calculator
- You have sample data and want to estimate a population mean.
- You are measuring a proportion (success/failure outcomes) and want a CI for the true proportion.
- You need a quick, transparent way to compare precision across different confidence levels like 90%, 95%, or 99%.
Inputs You Can Provide
- Sample size (n)
- For mean CI: sample mean and either the sample standard deviation (s) or known population standard deviation (?)
- For proportion CI: number of successes out of n
- Confidence level (e.g., 95%)
- Distribution: Z (when ? is known or n is large) or t (when ? is unknown and n is small)
How the Calculator Works
Mean Confidence Interval
For a mean, the general form is:
CI = x̄ ± critical × (SD / ?n)
- Use the Z critical value when the population standard deviation ? is known (or when n is large and normality is reasonable).
- Use the t critical value with degrees of freedom n?1 when ? is unknown and n is small; this accounts for additional uncertainty from estimating ? with s.
Proportion Confidence Interval
For a proportion, the calculator uses the commonly taught Wald interval:
CI = p̂ ± z × ?[p̂(1?p̂)/n]
where p̂ = successes / n. This is fast and easy to interpret. For very small samples or extreme proportions near 0 or 1, more advanced intervals (like Wilson) may be preferred. For most moderate sample sizes, the classic Wald CI is a clear starting point.
Choosing the Right Confidence Level
Common choices are 90%, 95%, and 99%. Higher confidence means wider intervals; lower confidence yields narrower intervals. Select a level that balances certainty and precision for your decision context.
Interpreting Your Results
- If a 95% CI for a mean is [48.2, 51.6], it suggests the true population mean is plausibly between 48.2 and 51.6.
- If a 95% CI for a proportion is [0.42, 0.55], you can say the true proportion likely lies between 42% and 55%.
- Overlapping CIs between two groups do not necessarily mean no difference; formal hypothesis tests may be needed.
Best Practices
- Inspect data quality and outliers before interpreting intervals.
- For means with small n and unknown ?, prefer the t distribution over Z.
- Ensure the sample is random and representative of the population of interest.
- Report the point estimate, the confidence level, and the full interval bounds.
Example Use Cases
- Quality control: Estimate the true average measurement of a process output.
- Marketing: Estimate the conversion rate (proportion of successes) from an A/B test.
- Healthcare: Estimate average treatment effects or response proportions.
Why Use This Confidence Interval Calculator?
It’s purpose-built for clarity: you supply only the inputs that matter, and it returns interpretable results with the critical value used, the margin of error, and the final bounds. Adjust the decimal places to match your reporting standards and export the results directly to your workflow.