Use this Confidence Interval Calculator to estimate a confidence interval for a population mean or population proportion from sample data.

Calculation Method

This calculator currently uses Z-based confidence intervals.

Mean confidence interval:

x̄ ± z × (s ÷ √n)

Proportion confidence interval:

p̂ ± z × √[p̂(1 − p̂) ÷ n]

Where:

p̂ = number of successes ÷ sample size

Example

Suppose a sample has:

Mean = 50
Standard deviation = 10
Sample size = 100
Confidence level = 95%

The standard error is:

SE = 10 ÷ √100 = 1

Using z = 1.96:

Margin of error = 1.96 × 1 = 1.96

The 95% confidence interval is:

50 ± 1.96 = 48.04 to 51.96

Important Note

A 95% confidence interval does not mean there is a 95% probability that the true value lies inside one calculated interval. Under repeated sampling with the same method, approximately 95% of such intervals would contain the true parameter.

For small samples with an unknown population standard deviation, a t-based mean interval may be more appropriate. Simple normal proportion intervals may also be unsuitable for small samples or proportions close to 0 or 1.

Sources

• NIST/SEMATECH — What Are Confidence Intervals?
• OpenStax Introductory Statistics 2e — Confidence Intervals