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Confidence Interval Calculator

Find a two-sided z confidence interval for a mean when the population standard deviation is known.

Calculate

Known population σ only: enter a sample mean, positive population SD and integer sample size. Sample-SD t intervals and proportion intervals are not supported.

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Formula and variables

CI = x̄ ± z* σ/√n

x̄ is the observed sample mean, σ the known population SD, n the number of independent observations and z* the two-sided standard-normal critical value.

Worked example

For x̄ = 50, σ = 10, n = 100 and 95% confidence: SE = 10/√100 = 1. The critical value is about 1.959964. Margin = 1.959964 × 1, giving [48.040036, 51.959964].

When to use this formula

Use this interval for an independent random sample from a normal population with known σ, or when an appropriate large-sample approximation is justified. A sample SD is not known population σ; use a t procedure for that case.

Common mistakes

A 95% confidence interval does not assign 95% probability to the fixed population mean after the data are observed. In repeated sampling, this procedure covers the true mean in 95% of intervals under its assumptions.

Can I enter my sample standard deviation here?

No. That substitutes an estimate for a known parameter and requires a different critical value. See the confidence-interval guide for choosing a t interval.

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