Confidence Interval Calculator
Use this tool to calculate a confidence interval for a population mean (known standard deviation) or a population proportion.
What is a confidence interval?
A confidence interval is a range of plausible values for an unknown population parameter. Instead of giving a single estimate, it gives you an interval with an associated confidence level such as 95%.
In practical terms, a 95% confidence interval means that if you repeated the same sampling process many times and built intervals the same way each time, about 95% of those intervals would contain the true parameter.
How this confidence level interval calculator works
This page supports two common z-based intervals:
- Mean (σ known): Use when the population standard deviation is known.
- Proportion: Use when estimating a percentage, rate, or probability from sample counts.
Mean CI: x̄ ± z* × (σ / √n)
Proportion CI: p̂ ± z* × √(p̂(1-p̂)/n)
where z* is the critical value based on your confidence level.
Input guide
- Confidence Level (%): The desired confidence (for example, 95).
- Sample Mean (x̄): Average from your sample data.
- Population Standard Deviation (σ): Known spread for mean intervals.
- Sample Size (n): Number of observations in your sample.
- Successes (x): Count of “yes” outcomes for proportion intervals.
Example scenarios
Example 1: Mean interval
Suppose test scores have a known population standard deviation of 12. You sample 64 students and get a mean of 78. At 95% confidence, the tool calculates:
- Critical value z* ≈ 1.96
- Margin of error ≈ 1.96 × (12/8) = 2.94
- Interval ≈ [75.06, 80.94]
Example 2: Proportion interval
If 52 out of 100 users prefer a new design, then p̂ = 0.52. At 95% confidence, this gives an interval around that estimate using the standard error for proportions.
Common mistakes to avoid
- Confusing confidence level with probability that a single computed interval is true.
- Using too small a sample size and expecting narrow intervals.
- Entering percentages as whole numbers in the wrong fields (for proportion, use counts: x and n).
- Using z-interval assumptions in settings where they may not hold.
Interpretation tips
A wider interval means more uncertainty. You can reduce width by increasing sample size. Choosing a higher confidence level (like 99% instead of 95%) increases certainty but also widens the interval.
For business, science, and product analytics, confidence intervals communicate both estimate and reliability, making decisions more transparent than a single-point estimate alone.