z interval for a mean (σ known)

When the population standard deviation σ is known, the interval for μ is x̄ ± z·σ/√n.

\xbar \pm z_{1-\alpha/2}\,\frac{\sigma}{\sqrt{n}}

Two-sided z interval

$\xbar$: sample mean
$\sigma$: known population standard deviation
$z_{1-\alpha/2}$: standard normal quantile: 1.645 (90%), 1.960 (95%), 2.576 (99%)
$n$: sample size

When to use

Rare in practice: σ is known from long process history or a well-established measurement system. It is mostly a teaching step before the t interval. For one-sided alternatives the engine returns the matching one-sided bound using $z_{1-\alpha}$ .

Assumptions

Random sample, independent observations, a normal population or $n$ large enough for the CLT, and σ genuinely known.

How to interpret

Margin of error $= z\,\sigma/\sqrt{n}$ . Halving it needs four times the sample size.

Common mistakes

Plugging the sample SD into the z formula with a small sample (use t).
Using $z_{0.95} = 1.645$ for a two-sided 95% interval.

Example

$\xbar = 52.3$ , $\sigma = 6$ , $n = 36$ . Margin $= 1.96 \times 6/6 = 1.96$ ; the 95% interval is $(50.34, 54.26)$ .