Confidence Interval Calculator

A confidence interval (CI) is a range of values that is likely to contain the true population parameter (such as the population mean) with a specified level of confidence. For example, a 95% confidence interval means that if you repeated the sampling process 100 times, approximately 95 of the resulting intervals would contain the true population mean. Our confidence interval calculator computes the CI for a population mean from sample statistics: sample mean (x̄), standard deviation (s), and sample size (n). The formula is: CI = x̄ ± (critical value × SE), where SE = s / √n is the standard error.

1. 1. Enter Sample Statistics: Input the sample mean (x̄), standard deviation (s), and sample size (n). Select your confidence level (90%, 95%, 99%, 99.9%, or custom) and distribution type. All processing happens locally in your browser — your data never leaves your device.
2. 2. Critical Value Computed: The confidence interval calculator finds the critical value from the selected distribution. For the t-distribution, it uses the degrees of freedom df = n − 1. For the z-distribution, it uses the standard normal table. The critical value corresponds to the tail probability α/2 = (1 − confidence level) / 2.
3. 3. CI Bounds Displayed: The confidence interval calculator computes the margin of error (ME = critical value × SE) and the CI bounds (lower = x̄ − ME, upper = x̄ + ME). Results include a visual bar, step-by-step breakdown, and a plain-language interpretation.

• Use t-distribution when: The population standard deviation (σ) is unknown and you are estimating it from the sample (using s). This is the most common case in practice. The t-distribution has heavier tails than the normal distribution, producing wider CIs that account for the additional uncertainty from estimating σ. Use t when n < 30, or whenever σ is unknown regardless of sample size. Our confidence interval calculator uses df = n − 1 degrees of freedom.
• Use z-distribution when: The population standard deviation (σ) is known exactly, or when n is large (n ≥ 30) and the Central Limit Theorem ensures the sampling distribution is approximately normal. Common z critical values: z* = 1.645 (90%), 1.960 (95%), 2.576 (99%).
• Practical note:For n ≥ 30, the t and z distributions give very similar results. For n < 30, the t-distribution gives wider (more conservative) intervals that are statistically more appropriate. Our confidence interval calculator defaults to the t-distribution as the safer choice.
• Confidence level interpretation: A 95% CI does NOT mean there is a 95% probability that the true mean is in this specific interval. It means that the procedure used to construct the interval will capture the true mean 95% of the time across repeated samples. The true mean is either in the interval or it is not — the probability refers to the long-run frequency of the method.

• Sample size (n): Larger samples produce narrower CIs because SE = s / √n decreases as n increases. Doubling the sample size reduces the CI width by a factor of √2 ≈ 1.41.
• Standard deviation (s): Higher variability in the data produces wider CIs. Reducing measurement noise or controlling experimental conditions narrows the CI.
• Confidence level: Higher confidence levels require wider intervals. A 99% CI is always wider than a 95% CI for the same data, because you need to cast a wider net to be more confident.