stats

Variance & Std Error

Population vs sample variance, SEM, confidence intervals, and how SEM shrinks as n grows.

Input — Data Set

Numbers (comma or space separated)

Variance

Variance measures how spread out values are from the mean.

Count n—

Mean x̄—

Population variance σ²—

Sample variance s²—

σ² uses ÷n (population); s² uses ÷(n−1) — Bessel's correction. s² is an unbiased estimator of the true population variance.

Standard Error

SEM estimates how far the sample mean is likely to be from the true mean.

SEM = s / √n—

90% CI x̄ ± 1.645·SEM—

95% CI x̄ ± 1.960·SEM—

99% CI x̄ ± 2.576·SEM—

CIs use Z-critical values. For n < 30, t-critical gives better coverage.

SEM = s/√n · CI = x̄ ± Z·SEM