How to use the confidence interval calculator
- Select whether you are estimating a mean or a proportion.
- Enter your sample statistic: the sample mean or the observed proportion (between 0 and 1).
- Provide the sample size (must be 2 or greater) and, for means, the standard deviation.
- Choose a confidence level from the dropdown menu.
- Read the computed margin of error and the lower and upper bounds of the interval.
How the confidence interval is calculated
The formula depends on whether you are estimating a population mean or a population proportion. Both rely on the normal approximation, which works well when the sample size is large enough.
For a mean, the interval is x̄ ± z × (s / √n), where x̄ is the sample mean, s is the standard deviation, n is the sample size, and z is the critical value for your chosen confidence level. For example, with a mean of 100, a standard deviation of 15, and a sample size of 25 at 95% confidence (z = 1.96): the standard error is 15 / √25 = 3, the margin of error is 1.96 × 3 = 5.88, and the interval spans from 94.12 to 105.88.
For a proportion, the formula is p̂ ± z × √(p̂(1 - p̂) / n). The standard error uses the sample proportion itself because the variance of a binary outcome is determined entirely by p̂.
Critical values for common confidence levels
This confidence interval calculator uses z-scores from the standard normal distribution. Here are the critical values for the three most frequently used confidence levels:
| Confidence level | Critical value (z) |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
A higher confidence level requires a larger critical value, which widens the interval. Choosing 99% instead of 95% gives you more certainty that the interval contains the true parameter, but the trade-off is a less precise estimate.
When to use this approach
The normal approximation used here is appropriate when certain conditions are met. For means, the central limit theorem ensures the sampling distribution is roughly normal when the sample size is around 30 or more, even if the underlying population is skewed. For smaller samples drawn from a normal population, a t-distribution would technically be more accurate, but the z-based method remains a practical standard for quick estimates.
For proportions, the rule of thumb is that both n × p̂ and n × (1 - p̂) should be at least 10. If your sample has very few successes or failures, the normal approximation breaks down and exact methods like the Clopper-Pearson interval are preferred.
Common mistakes to avoid
One frequent error is confusing the confidence level with probability. A 95% confidence interval does not mean there is a 95% chance the true parameter lies inside your specific interval. It means that if you repeated the sampling process many times, about 95% of the resulting intervals would contain the true value.
Another mistake is entering percentages instead of decimals for proportions. If 62 out of 100 people answered yes, enter 0.62, not 62. Entering a number outside the 0 to 1 range will produce an invalid result.
Finally, remember that the standard deviation field applies only to mean calculations. When computing a proportion interval, the variability is derived from the proportion itself, so the standard deviation input is ignored.
Relationship to other statistical tools
Confidence intervals are closely related to hypothesis testing. If a 95% confidence interval for a mean difference does not include zero, the corresponding two-sided test would reject the null hypothesis at the 0.05 significance level. This duality makes intervals more informative than isolated test statistics because they show the range of plausible values rather than just a pass or fail decision.
While this tool handles single-sample estimation, researchers often need to compare groups or model relationships. Those tasks involve different methods and dedicated calculators for regression or matrix operations.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Mean height of students | Parameter type: mean; Sample mean or proportion (p̂): 170; Sample size (n): 50; Standard deviation (s or σ): 8; Confidence level: 95 | Margin of error: 2.217487; Lower bound: 167.782513; Upper bound: 172.217487; Standard error: 1.131371; Critical value (z): 1.96 |
| Survey approval proportion | Parameter type: proportion; Sample mean or proportion (p̂): 0.62; Sample size (n): 400; Standard deviation (s or σ): 0; Confidence level: 95 | Margin of error: 0.047568; Lower bound: 0.572432; Upper bound: 0.667568; Standard error: 0.024269; Critical value (z): 1.96 |
| Test scores at 99% confidence | Parameter type: mean; Sample mean or proportion (p̂): 75; Sample size (n): 30; Standard deviation (s or σ): 10; Confidence level: 99 | Margin of error: 4.703111; Lower bound: 70.296889; Upper bound: 79.703111; Standard error: 1.825742; Critical value (z): 2.576 |
FAQ
What is a confidence interval?
A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter. The width of the range reflects the uncertainty inherent in sampling.
Why does a higher confidence level produce a wider interval?
To be more certain that the interval captures the true parameter, you must allow for more variability. Mathematically, a higher confidence level uses a larger z-score, which increases the margin of error.
Can I use this confidence interval calculator for small samples?
For means, the normal approximation works best with 30 or more observations. For smaller samples from a normally distributed population, a t-based interval is more accurate, though the z-based result is still commonly used as an estimate.
Should I enter a proportion as a percentage or a decimal?
Enter it as a decimal between 0 and 1. For example, if 45% of respondents chose an option, type 0.45.
Is my data sent to a server?
No. All calculations happen locally in your browser. Nothing you type is uploaded or stored anywhere.
What is the margin of error?
The margin of error is half the width of the confidence interval. It represents the maximum expected difference between the sample statistic and the true population parameter at the chosen confidence level.