Sigma

How to read it

Dictionary.com gives the pronunciation SIG-muh /ˈsɪg mə/. Sigma is the Greek S, and it has three forms: the capital Σ, the small σ, and ς, which the Collins English Dictionary says is used at the end of a word. Unicode records that Modern Greek calls the final form “stigma”, a name it warns is not to be confused with the old letter stigma (Ϛ).

Also searched as: final sigma ς, sigma letter, sigma symbol, σ, Σ, σ meaning, Σ meaning.

Capital and small forms

The summation sign ∑ is a separate character from the letter Σ, though Unicode cross-references the two, and the Latin letter esh Ʃ is another look-alike.

Sigma in maths and statistics

The Collins English Dictionary defines Σ in maths as the symbol indicating summation of the numbers or quantities that follow; OpenStax labels the sum of a column “SUM (Σ)”. In statistics σ is the population standard deviation and σ² the population variance; the standard deviation of a sample is written s (OpenStax Introductory Statistics). The population mean is written μ, as in OpenStax’s x = μ + (number of standard deviations)(σ).

In AI and machine learning

σ is the sigmoid function, σ(x) = 1 / (1 + exp(−x)), which PyTorch’s documentation writes exactly so for its Sigmoid layer. Σ is a covariance matrix: scikit-learn’s guide to covariance estimation writes a shrunk covariance matrix as Σ_shrunk.

In quantum physics

σ₁, σ₂ and σ₃ are the Pauli matrices. MIT’s 8.05 notes by Barton Zwiebach, on the spin of the electron, define the spin operators through them and note that Pauli himself developed much of the mathematics of spin one-half in 1927.

Other uses in physics

σ is the Stefan–Boltzmann constant, exactly 5.670 374 419 × 10⁻⁸ W m⁻² K⁻⁴ (CODATA 2022, NIST), and surface charge density, σ = Q / A (NIST’s Guide for the Use of the SI). Σ is the sigma baryon, a particle of three quarks that OpenStax lists at 1189.4 MeV.