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.:

$${\displaystyle \sum _{k=0}^{5}k=0+1+2+3+4+5=15}$$

• In mathematical logic, ${\displaystyle \Sigma _{n}^{0}}$ is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating ${\displaystyle n-1}$ times between existential and universal quantifiers. This notation reflects an indirect analogy between the relationship of summation and products on one hand, and existential and universal quantifiers on the other. See the article on the arithmetic hierarchy.
• In statistics, σ represents the standard deviation of population or probability distribution (where mu or μ is used for the 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, sometimes in the form ${\displaystyle \;|\!\!\!\Sigma }$ to distinguish it from the summation operator.
• Theoretical spectral analysis uses σ as standard deviation opposed to lowercase mu as the absolute mean value.

Biology, physiology, and 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^[10]

Business, finance, and 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.
• In alchemy, Σ was sometimes used to represent sugar.

Engineering and computer science

• In computer science, Σ represents the set of symbols that form an alphabet (see also linguistics)
• Relational algebra uses the values ${\displaystyle \sigma _{a\theta b}(R)}$ and ${\displaystyle \sigma _{a\theta v}(R)}$ 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.
• In civil engineering, σ refers to the normal stress applied on a material or structure.

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 astronomy, the prefix Σ is used to designate double stars of the Catalogus Novus Stellarum Duplicium by Friedrich Georg Wilhelm von Struve.
• In particle physics, Σ represents a class of baryons.