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Sigma (Σ, σ) Definition

Sigma (Σ, σ) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, note the two lowercase sigmas (σ) in the center of the name and the word-final sigma (ς) at the end.

Usage in Mathematics & Science

The lowercase (σ) & uppercase (Σ) sigma are used for:

  • Language & Linguistics:

    • In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative /s/. In Modern Greek, this sound is voiced to /z/ before /m/, /n/, /v/, /ð/ or /ɣ/.

    • The uppercase form of sigma (Σ) was re-borrowed into the Latin alphabet—more precisely, the International African Alphabet—to serve as the uppercase of modern esh (lowercase: ʃ).

    • In phonology, σ is used to represent syllables.

    • In linguistics, Σ represents the set of symbols that form an alphabet (see also computer science).

  • Science & Mathematics:

    • In general mathematics, lowercase σ is commonly used to represent unknown angles, as well as serving as a shorthand for "countably", whereas Σ is regularly used as the operator for summation, e.g.:

    • In statistics, σ represents standard deviation of population or probability distribution (as opposed to lowercase mu or μ, used for mean).

    • In topology, σ-compact topological space is one that can be written as a countable union of compact subsets.

    • In mathematical analysis and in probability theory, there is a type of algebra of sets known as σ-algebra (aka σ-field). Sigma algebra also includes terms such as:

      • σ(A), denoting the generated sigma-algebra of a set A

      • σ-finite measure (see measure theory)

    • In number theory, σ is included in various divisor functions, especially the sigma function or sum-of-divisors function.

    • In applied mathematics, σ(T) denotes the spectrum of a linear map T.

    • In complex analysis, σ is used in the Weierstrass sigma-function.

    • In probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, to distinguish it from the summation operator.

    • In statistics (as mentioned above), σ represents the standard deviation of a population or probability distribution.

    • Theoretical spectral analysis uses σ as standard deviation opposed to lowercase mu as the absolute mean value.

  • Biology, Physiology, & Medicine:

    • In biology, the sigma receptor (σ–receptors) is a type of cell surface receptor.

    • In biochemistry, the σ factor (or specificity factor) is a protein found in RNA polymerase.

    • In bone physiology, the bone remodeling period—i.e., the life span of a basic multicellular unit—has historically been referred to as the sigma period.

    • In early 20th-century physiology literature, σ had been used to represent milliseconds.

  • Business, Finance, & Economics:

    • In finance, σ is the symbol used to represent volatility of stocks, usually measured by the standard deviation of logarithmic returns.

    • In accounting, Σ indicates the balance of invoice classes and the overall amount of debts and demands.

    • In macroeconomics, σ is used in equations to represent the elasticity of substitution between two inputs.

    • In the machine industry, Six Sigma (6σ) is a quality model based on the standard deviation.

  • Chemistry:

    • Sigma bonds (σ bonds) are the strongest type of covalent chemical bond.

    • In organic chemistry, σ symbolizes the sigma constant of Hammett equation.

  • Engineering & Computer Science:

    • In computer science, Σ represents the set of symbols that form an alphabet (see also linguistics)

    • Relational algebra uses the values σaθb(R) and σaθv(R) to denote selections, which are a type of unary operation.

    • In machine learning, σ is used in the formula that derives the Sigmoid function.

    • In radar jamming or electronic warfare, radar cross-sections (RCS) are commonly represented as σ when measuring the size of a target's image on radar.

    • In signal processing, σ denotes the damping ratio of a system parameter.

    • In theoretical computer science, Σ serves as the busy beaver function.

  • Physics:

    • In nuclear and particle physics, σ is used to denote cross sections in general(see also RCS), while Σ represents macroscopic cross sections [1/length].

    • The symbol is to denote the Stefan–Boltzmann constant.

    • In relation to fundamental properties of material, σ is often used to signify electrical conductivity.

    • In electrostatics, σ represents surface charge density.

    • In continuum mechanics, σ is used to signify stress.

    • In condensed matter physics, Σ denotes self-energy.

    • The symbol can be used to signify surface tension (alternatively, γ or T are also used instead).

    • In quantum mechanics, σ is used to indicate Pauli matrices.

    • In astronomy, σ represents velocity dispersion

    • In particle physics, Σ represents a class of baryons.

  • Organizations:

    • During the 1930s, an uppercase Σ was in use as the symbol of the Ação Integralista Brasileira, a fascist political party in Brazil.

    • Sigma Corporation uses the name of the letter but not the letter itself, but in many Internet forums, photographers refer to the company or its lenses using the letter.

    • Sigma Aldrich incorporate both the name and the character in their logo.

Greek Alphabet

The letters of the Ancient Greek Alphabet, which are frequently utilized in math and science:

Greek Alphabet

Symbol

Letter

Symbol

Letter

Uppercase

Lowercase

Uppercase

Lowercase

Α

α

Alpha

Ν

ν

Nu

Β

β

Beta

Ξ

ξ

Xi

Γ

γ

Gamma

Ο

ο

Omicron

Δ

δ

Delta

Π

π

Pi

Ε

ε

Epsilon

Ρ

ρ

Rho

Ζ

ζ

Zeta

Σ

σ

Sigma

Η

η

Eta

Τ

τ

Tau

Θ

θ

Theta

Υ

υ

Upsilon

Ι

ι

Iota

Φ

φ

Phi

Κ

κ

Kappa

Χ

χ

Chi

Λ

λ

Lambda

Ψ

ψ

Psi

Μ

μ

Mu

Ω

ω

Omega

Related Definitions

Sources

“Sigma.” Wikipedia, Wikimedia Foundation, 6 May 2020, en.wikipedia.org/wiki/Sigma.

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