\sigma

Sigma (lowercase)

Naming convention: meaning depends on the field

In plain English

Usually the standard deviation: the typical distance from the mean. $\sigma^2$ is the variance. "Five sigma" in particle physics means a result about five standard deviations from what chance alone would produce. In ML, $\sigma(\cdot)$ written as a function means the sigmoid $1/(1+e^{-x})$ or a generic nonlinearity. Do not confuse it with capital $\Sigma$ (sum), or with $\Sigma$ as a covariance matrix.

As code

const xs = [2, 4, 4, 5, 7];
const mu = 4.4;
const sigma = Math.sqrt(xs.reduce((a, x) => a + (x - mu) ** 2, 0) / xs.length);

See it

μ (mean)0.00σ (std. dev.)1.10

\mu

shifts the centre and

\sigma

sets the spread. About 68% of the probability lies within one σ of μ.

What it means, field by field

Mathematics

Standard deviation; σ-algebras in measure theory; singular values σ_i in the SVD.

Physics

Statistical significance (“5σ”); cross-sections; Stefan–Boltzmann constant; conductivity.

AI / ML

Sigmoid or generic activation function σ(x); noise level in diffusion models; singular values of weight matrices.

Biology

Variation in a trait; spread of a dose-response; noise in gene expression.

In a research paper

The normal (Gaussian) distribution: measurement error, the central limit theorem, weight initialisation.

p(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}

A bell curve centred at μ whose width is set by σ; the front factor only makes the total probability 1.

Common misreadings

Mixing up σ (spread) and Σ (sum or covariance matrix).
Reading σ(x) as "sigma times x" when it denotes a function (sigmoid).
Treating "5σ" as "a 5% chance of being wrong". It is a far smaller tail probability, and it still says nothing about systematic error.

Connected symbols

\mu

\sum

Summation (capital sigma)

\mathbb{E}

Expectation (expected value)

e^{x}

Exponential

Where it appears on this site

§ 4. Probability, Entropy, and Information § 8. The Scientific Skeptic’s Toolkit