Sigma Symbol (Σ σ ς): Meaning, Standard Deviation, and How to Type It

Sigma is the eighteenth letter of the Greek alphabet, and it is probably the Greek letter you will meet most often outside a Greek classroom. The uppercase Σ tells you to add things up; the lowercase σ is the standard deviation symbol in statistics and the stress symbol in engineering.

This guide explains what each form means, works through summation notation and the standard deviation formula, and shows how to type Σ, σ, and ς on Windows, Mac, iPhone, in HTML, and in LaTeX.

Sigma Symbol Copy and Paste

Need the character right now? Select and copy any of these:

• Σ — uppercase sigma (summation symbol)
• σ — lowercase sigma (standard deviation symbol)
• ς — final sigma (used at the end of Greek words)
• σ² — variance

The Sigma Symbol Explained

Sigma has three shapes in modern Greek:

• Σ — uppercase sigma, Unicode U+03A3
• σ — lowercase sigma, Unicode U+03C3
• ς — lowercase final sigma, Unicode U+03C2

The name is pronounced "SIG-muh." In the Greek numeral system sigma stood for 200, and its sound is the plain "s" that the Latin letter S inherited.

Uppercase Σ vs Lowercase σ and Final Sigma ς

The three forms are not interchangeable, and choosing the wrong one changes the meaning of an equation.

Uppercase Σ (U+03A3) is the summation symbol. In math and statistics it is an operator, not a variable: it tells the reader to add every term that follows it.

Lowercase σ (U+03C3) is a variable. Depending on the field it stands for standard deviation, mechanical stress, electrical conductivity, surface charge density, or a scattering cross-section.

Final sigma ς (U+03C2) is a spelling rule from the Greek language. When a lowercase sigma falls at the end of a word, Greek writes it as ς instead of σ. The word for "wisdom," σοφία, uses the normal form, while "Odysseus," Οδυσσεύς, ends in the final form. You will almost never see ς in an equation, so if you need sigma as a variable, you want σ.

One look-alike to watch for: ∑ (U+2211), produced by Option + W on a Mac or ∑ in HTML, is a math operator with its own code point, not the letter Σ.

Summation Notation (Σ)

Sigma notation compresses a long sum into a compact expression:

Σ (i = 1 to n) xᵢ

Read it as "the sum of x sub i, for i running from 1 to n." It unpacks to x₁ + x₂ + x₃ + … + xₙ.

Parts of a summation: the Σ itself; the index variable (usually i, j, or k) and its lower limit below the Σ; the upper limit above it; and the summand to the right, evaluated once for each index value.

Worked Example

Take the sum of i² for i from 1 to 5:

Σ (i = 1 to 5) i² = 1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55

A second example with a closed-form answer is the sum of the first n integers:

Σ (i = 1 to n) i = n(n + 1) / 2

For n = 10 this gives 10 × 11 / 2 = 55, which you can confirm by adding 1 through 10 by hand.

Infinite series use ∞ as the upper limit, for example Σ (n = 0 to ∞) 1/2ⁿ = 2.

Standard Deviation Symbol (σ)

In statistics, σ is the symbol for population standard deviation, the most widely used measure of how far data points spread out from the mean. A small σ means the values cluster tightly around the average; a large σ means they are scattered.

The Population Standard Deviation Formula

σ = √[ Σ (xᵢ − μ)² / N ]

where xᵢ is each value, μ (mu) is the population mean, N is the number of values, and Σ adds the squared deviations across all N values. Notice that the formula uses both sigmas: uppercase Σ to add, lowercase σ as the name of the result.

Example: For the data set 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Dividing by N = 8 gives 4, and the square root gives σ = 2.

σ vs s: Population vs Sample

Dividing by n − 1 (Bessel's correction) makes s² an unbiased estimate of σ². For the same eight numbers, s is about 2.14 rather than 2.

Variance: σ²

Variance is the square of the standard deviation, which is why it is written σ² ("sigma squared"). Variance is convenient in algebra because the variances of independent variables add directly. Standard deviation is easier to interpret because it has the same units as the data.

The Empirical Rule (68-95-99.7)

For a normal distribution, σ gives a quick picture of where the data lie:

• About 68.3% of values fall within 1σ of the mean
• About 95.4% fall within 2σ
• About 99.7% fall within 3σ

This rule is why scientists describe results in "sigmas." A measurement that sits 5σ from the null prediction has roughly a one-in-3.5-million chance of being a fluke, the threshold particle physicists use to claim a discovery.

Six Sigma

Six Sigma is a quality-management methodology developed at Motorola in the 1980s. The name refers to a process whose specification limits lie six standard deviations from the mean; allowing for a 1.5σ long-term drift, that works out to about 3.4 defects per million opportunities.

Sigma in Physics, Chemistry, and Astronomy

Physics and Engineering

• Stress — σ = F/A, force per unit area, in pascals. Normal stress is σ, shear stress is τ, and the full stress tensor is σᵢⱼ.
• Electrical conductivity — σ = 1/ρ, the reciprocal of resistivity, in siemens per metre.
• Stefan–Boltzmann constant — σ ≈ 5.670 × 10⁻⁸ W·m⁻²·K⁻⁴, giving the power radiated by a black body: P = σAT⁴.
• Cross-section — in nuclear and particle physics, the effective target area for a collision, measured in barns.

Chemistry

A sigma bond (σ bond) is a covalent bond formed by head-on overlap of atomic orbitals along the axis joining two nuclei. Every single bond is a sigma bond; double and triple bonds consist of one σ bond plus one or two π bonds. Sigma bonds are stronger than pi bonds and allow free rotation around the bond axis.

Astronomy

Astronomers use σ for velocity dispersion, the spread of velocities among stars in a galaxy or galaxies in a cluster. It is a direct probe of mass, and the M–σ relation ties a galaxy's central black hole mass to the velocity dispersion of its bulge.

Mathematics

• Divisor function — σ(n) is the sum of the positive divisors of n, so σ(6) = 12
• Sigma-algebra — in measure theory, a family of sets closed under countable unions and complements

How to Type the Sigma Symbol

Windows: Hold Alt and type 228 on the numeric keypad for Σ, or Alt + 229 for σ. These codes come from the old DOS code page 437, so they work almost everywhere. The four-digit codes (Alt + 931, Alt + 963) only work in programs that accept Unicode alt codes, such as Word, where typing 03C3 and pressing Alt + X is the most reliable method.

Mac: The US keyboard has no Option shortcut for Σ or σ; Option + W produces ∑, the summation operator. For the real letters, add Greek under System Settings > Keyboard > Input Sources, then press S for σ, Shift + S for Σ, and W for ς.

iPhone, iPad, and Android: Add the Greek keyboard in your keyboard settings, switch with the globe key, and tap S for σ or Shift + S for Σ.

Sigma in LaTeX

In math mode, \Sigma gives Σ, \sigma gives σ, \varsigma gives ς, and \sum gives the large summation operator, which is what you want for actual sums. Limits attach with _ and ^:

1\sum_{i=1}^{n} x_i
2\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2}
3\sum_{n=0}^{\infty} \frac{1}{2^n} = 2
Copy

In display math the limits sit above and below the Σ; inline they move to the side (\sum\limits forces the stacked layout). Reserve \Sigma for sigma as a variable name, such as a covariance matrix, never for a sum.

Common Sigma Expressions

• Σxᵢ — sum of all x values
• Σ(xᵢ − μ)² — sum of squared deviations from the mean
• σ = √[Σ(xᵢ − μ)²/N] — population standard deviation
• σ² — variance
• σ = F/A — mechanical stress

Frequently Asked Questions

What does the sigma symbol mean?

Uppercase Σ means "sum of" and tells you to add a sequence of terms. Lowercase σ is a variable that most often stands for standard deviation in statistics; in physics it can mean stress, conductivity, or a cross-section.

What is the difference between σ and s?

σ is the standard deviation of a whole population (dividing by N); s is the standard deviation of a sample (dividing by n − 1) and estimates σ.

Why does Greek have two lowercase sigmas?

σ is used at the beginning or middle of a word and ς at the end. It is a handwriting convention that survived into print. Formulas use only σ.

How do I type sigma on a keyboard?

On Windows, hold Alt and type 228 for Σ or 229 for σ on the numeric keypad. On a Mac, switch to the Greek input source and press S or Shift + S. In LaTeX, write \sigma or \Sigma in math mode, and \sum for sums.

Writing with Greek Symbols

Statistical reports and physics papers lean on sigma in both cases, often on the same page. In LaTeX, keeping sums as \sum and variables as \sigma gives correct spacing and italics automatically.

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