The standard deviation symbol: σ, s, and everything around them

The complete notation table

The rule behind the whole system

Statistics uses a consistent convention that makes almost all of this table predictable:

• Greek letters describe populations. σ, μ, ρ, β. These are parameters — fixed, real, and usually unknowable.
• Roman letters describe samples. s, x̄, r, b. These are statistics — computed from data you actually have, and different for every sample you might draw.

So the question "σ or s?" is really the question "is this number describing a whole population, or a sample drawn from one?" — the same question that decides which denominator to use in the formula. Answer it once and the notation follows.

σ and Σ are not the same symbol

Both are called sigma and they appear in the same formula, which causes real confusion:

σ = √[ Σ(x − μ)² / N ]

Lowercase σ is a quantity — the standard deviation itself, the thing being defined. Capital Σ is an instruction — an operator that says "sum the following over every value". One is a noun, the other is a verb.

How to type each symbol

In Word, typing the Unicode hex code and pressing Alt+X converts it in place — so 03C3 then Alt+X gives σ. On a Mac, Ctrl+Cmd+Space opens the character viewer where you can search by name.

Note that x̄ is not a single character: it is the letter x followed by U+0304, the combining macron. That is why it sometimes renders badly in fonts that lack proper combining-mark support, and why copying it out of a PDF often loses the bar.

Sx or σx? Reading the symbols on your calculator

Run one-variable statistics on any calculator and it returns two standard deviations, not one. Neither is wrong and neither is a rounding of the other — they are the sample and population figures, sitting next to each other, and the machine will not tell you which one your assignment wants.

On a TI, Sx is the sample standard deviation (the n − 1 one) and σx is the population standard deviation (the N one). If you are working from a sample — which, in a statistics class, you almost always are — the answer you want is Sx.

Two traps worth knowing. The lowercase x in Sx and σx is just the name of the variable, not part of the statistic — on a list called L2 the same figures may appear as SL2 and σL2. And the σx a calculator prints is not the standard error, even though the standard error is also written σx̄; the bar over the x is the entire difference between them.

If the two numbers on your screen are far apart, your sample is small — that gap is Bessel's correction, and it shrinks to nothing as n grows. The TI-84 and Casio guides have the full keystrokes.

How to pronounce them out loud

Reading a formula aloud, σ = √[ Σ(x − μ)² / N ] is "sigma equals the square root of the sum of x minus mu, squared, over big N". Saying "sum over" for Σ rather than "sigma" keeps the operator and the quantity audibly distinct.

Copy and paste the symbols

Plain Unicode, safe to paste into Word, Google Docs, email or a spreadsheet cell:

The whole formulas, as one line each: σ = √(Σ(x − μ)² / N) and s = √(Σ(x − x̄)² / (n − 1)). The formula page has these in LaTeX, Excel and code form as well.

Notation you may also meet

σ̂ (sigma hat)

A hat over a symbol means "estimated from data", so σ̂ is an estimate of the population standard deviation σ. In most introductory texts it is simply the sample standard deviation s, though some authors use σ̂ for the version that divides by n instead of n − 1, so check which one a book means. In regression output, σ̂ usually denotes the residual standard error — the typical distance of the points from the fitted line. The same convention gives p̂ (an estimated proportion), μ̂ and β̂.

Other notation

• SD — plain text for standard deviation, common in journals: "mean 24.1 (SD 3.2)".
• SEM — standard error of the mean. Frequently confused with SD in published figures; check which one an error bar represents, because SEM is always the smaller and more flattering of the two.
• s_x and σ_x — the standard deviation of a variable named x, used when several variables are in play.