Back to Greek Alphabet
View Pronunciations | LaTeX Guide

Sigma (Σ, σ)

Σ Uppercase
σ Lowercase

Quick Info

  • Pronunciation: SIG-muh
  • English: s
  • Common Use: Sum, Standard deviation

Historical Background

Eighteenth letter of the Greek alphabet, derived from Phoenician letter "shin", meaning teeth. The symbol Σ has been used since ancient times to represent summation in mathematics, while σ is fundamental in statistics as the symbol for standard deviation.

Scientific Applications

Mathematics

Summation

∑i=1nxi\sum_{i=1}^n x_i

Statistics

Standard deviation

σ=∑(x−μ)2N\sigma = \sqrt{\frac{\sum(x-\mu)^2}{N}}

Physics

Surface charge density

σ=QA\sigma = \frac{Q}{A}

Materials Science

Stress

σ=FA\sigma = \frac{F}{A}

Chemistry

Molecular bonding

σ-bond\sigma\text{-bond}

Symbol Codes

Unicode

Uppercase

  • Code: U+03A3
  • Hex: Σ
  • Decimal: Σ

Lowercase

  • Code: U+03C3
  • Hex: σ
  • Decimal: σ

HTML

Uppercase

Σ

HTML entity for uppercase Sigma

Lowercase

σ

HTML entity for lowercase sigma

LaTeX

Uppercase

\Sigma

LaTeX command for uppercase Sigma

Lowercase

\sigma

LaTeX command for lowercase sigma

Common Usage

Mathematics

Series

Summation of sequences

Set Theory

Algebraic structures

Linear Algebra

Matrix operations

Statistics

Descriptive Statistics

Standard deviation

Inferential Statistics

Population variance

Data Analysis

Dispersion measures

Physics

Electrostatics

Surface charge density

Mechanics

Normal stress

Thermodynamics

Stefan-Boltzmann constant

Common Applications

Statistics

σ2=∑(xi−μ)2N\sigma^2 = \frac{\sum(x_i - \mu)^2}{N}

Population variance

s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}

Sample standard deviation

z=x−μσz = \frac{x - \mu}{\sigma}

Z-score calculation

Series and Sequences

∑n=1∞1n2=π26\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}

Basel problem

∑k=0n(nk)\sum_{k=0}^n \binom{n}{k}

Binomial expansion

∑r=0∞xr=11−x,∣x∣<1\sum_{r=0}^\infty x^r = \frac{1}{1-x}, |x| < 1

Geometric series

Special Applications

Engineering Stress

σ=Eϵ\sigma = E\epsilon

Hooke's law

σyield\sigma_{yield}

Yield stress

σultimate\sigma_{ultimate}

Ultimate stress

Statistical Analysis

CI=μ±zα/2σnCI = \mu \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}

Confidence interval

P(∣X−μ∣≤2σ)≈0.95P(|X-\mu| \leq 2\sigma) \approx 0.95

Empirical rule

σxˉ=σn\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}

Standard error

Summation Properties

Basic Properties

∑i=1n(axi+byi)=a∑i=1nxi+b∑i=1nyi\sum_{i=1}^n (ax_i + by_i) = a\sum_{i=1}^n x_i + b\sum_{i=1}^n y_i

Linearity

∑i=1nc=nc\sum_{i=1}^n c = nc

Constant sum

∑i=1ni=n(n+1)2\sum_{i=1}^n i = \frac{n(n+1)}{2}

Sum of integers

Advanced Properties

∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}

Sum of squares

∑i=1n∑j=1maij=∑j=1m∑i=1naij\sum_{i=1}^n \sum_{j=1}^m a_{ij} = \sum_{j=1}^m \sum_{i=1}^n a_{ij}

Double sum

∑k=0n(nk)=2n\sum_{k=0}^n \binom{n}{k} = 2^n

Binomial sum

Physical Applications

Materials Science

  • • Stress analysis
  • • Material strength
  • • Deformation studies
  • • Failure criteria

Statistical Physics

  • • Ensemble averages
  • • Fluctuation theory
  • • Distribution functions
  • • Partition functions

Quantum Mechanics

  • • Spin states
  • • Orbital notation
  • • Selection rules
  • • Coupling schemes

Theoretical Applications

Series Theory

∑n=1∞1ns\sum_{n=1}^\infty \frac{1}{n^s}

Riemann zeta function

∑n=0∞xnn!=ex\sum_{n=0}^\infty \frac{x^n}{n!} = e^x

Taylor series

∑n=0∞xn=11−x\sum_{n=0}^\infty x^n = \frac{1}{1-x}

Geometric series

Statistical Theory

σ2=E[(X−μ)2]\sigma^2 = E[(X-\mu)^2]

Variance definition

σ12+σ22=σtotal2\sigma_1^2 + \sigma_2^2 = \sigma_{total}^2

Variance addition

CV=σμCV = \frac{\sigma}{\mu}

Coefficient of variation

Writing Guidelines

Uppercase is a three-barred symbol (Σ) used for summation, lowercase looks like a curved o with a tail (σ). Note that there are two forms of lowercase sigma: σ (final form) and ς (word-end form). In handwriting, ensure the lowercase σ is clearly distinguished from the Latin o and the Greek omicron (ο).

How to Type Sigma

Windows

Alt Code

  • Hold Alt
  • Type 931 for Σ or 963 for σ
  • Release Alt

Character Map

  • Open Character Map
  • Select Greek
  • Find and copy Sigma

Unicode

  • Hold Alt + X
  • Type 03A3 for Σ or 03C3 for σ
  • Release Alt + X

macOS

Option Key

  • Press Option + w for σ

Character Viewer

  • Press Control + Command + Space
  • Select Greek
  • Click Sigma

Linux

Compose Key

  • Press Compose + s + s for σ

Unicode

  • Press Ctrl + Shift + U
  • Type 03C3
  • Press Enter

Frequently Asked Questions

What is the difference between Σ and ∑?

While both symbols represent summation, Σ is the Greek letter Sigma, while ∑ is the mathematical summation symbol. In practice, they are often used interchangeably in mathematical notation.

Why are there two forms of lowercase sigma (σ and ς)?

In Greek writing, ς (final sigma) is used when sigma appears at the end of a word, while σ is used in all other positions. In mathematical and scientific notation, σ is the standard form used regardless of position.

How is sigma used in statistics?

In statistics, σ (sigma) represents the standard deviation of a population, measuring the spread of data around the mean. The squared version, σ², represents variance.

What's the difference between σ and s in statistics?

σ (sigma) represents the population standard deviation, while s represents the sample standard deviation. σ is a parameter, while s is a statistic calculated from sample data.

Related Symbols

Statistical Measures

Mu (Mean)
Rho (Correlation)
Chi (Chi-square)

Mathematical Operators

Product (Product notation)
Integral (Integration)
Partial (Partial derivative)

Copied to clipboard