Standard Deviation Symbol

Two main symbols are used, depending on whether the data is a complete group or a sample:

• σ (lowercase Greek sigma) for population standard deviation.
• s for sample standard deviation.

That split mirrors the underlying difference: are you describing a complete group, or estimating a larger group from a slice of it? The same data, processed two different ways, can give two different numbers (which is why the symbols exist in the first place).

Notation can vary by textbook or software, but these are the most common meanings. The rest of this guide walks through every symbol you will see in standard deviation formulas.

The population standard deviation symbol is σ, the lowercase Greek letter sigma. You use σ when your data covers the entire group you are analyzing: every employee in a small company, every score in one classroom, every part in one production batch.

Note the capital version of the same Greek letter, Σ, is also used in standard deviation formulas, but it means something different. Capital Σ means “sum”: a shorthand for “add up everything that follows.” Lowercase σ is the result; capital Σ is one of the operations that produces it.

For more on when this version of the formula is the right choice, see Sample vs Population Standard Deviation.

The sample standard deviation symbol is s, the lowercase Roman letter. You use s when your data is a sample of a bigger group, and you want the result to estimate the spread of that bigger group: a survey of customers, a scientific study with limited samples, a window of recent measurements drawn from an ongoing process.

The convention is easy to remember: s for sample. In spreadsheet software the same idea shows up as a function name suffix. Excel's STDEV.S is sample standard deviation; STDEV.P is population. For the practical Excel walkthrough, see How to Calculate Standard Deviation in Excel.

The two symbols exist because there are two slightly different formulas behind them. Both formulas measure the same thing (spread around the mean), but they differ in one place:

• σ divides the sum of squared deviations by N (the total count).
• s divides by n − 1 (one less than the count). This is Bessel's correction.

Using the right symbol communicates which formula was used, which matters when you are comparing results across sources. Two reports of “standard deviation = 4” mean different things if one used σ and the other used s.

μ (lowercase Greek mu) is the population mean: the average of every value in the population. It is calculated by summing all the values and dividing by N (the total count).

μ shows up inside the population standard deviation formula because spread is measured relative to the mean. Each value's distance from μ is squared, then averaged, then square-rooted to give σ.

x̄ (read as “x-bar”) is the sample mean: the average of the values in a sample. It plays the same role in the sample standard deviation formula that μ plays in the population formula.

The calculation is identical: sum the sample values and divide by n (the sample count). Whether you call the result μ or x̄ depends on whether your data is the entire group or a sample. Some textbooks, calculators, and software just call it “mean” without distinguishing, then sort it out from context.

n (or its uppercase form N) is the count of values in the data set.

• N is usually used for the population count (every value in the entire group).
• n is usually used for the sample count (every value in the sample).

The capital-vs-lowercase distinction is a subtle visual cue about which formula is in play. Some sources use lowercase n for both and rely on context, but the capital-N convention shows up often enough that it is worth recognizing.

n − 1 (n minus one) is the denominator in the sample standard deviation formula. It looks like an off-by-one mistake at first, but the subtraction is intentional and useful.

A sample, by nature, is missing some of the values from the larger population. The values you happen to capture usually include fewer of the most extreme high or low values, so the spread you measure from a sample tends to be a little smaller than the true population spread. Dividing by n − 1 instead of n nudges the estimate up to compensate.

This adjustment is called Bessel's correction. It matters most when sample sizes are small and shrinks as sample size grows. With 5 values, dividing by 4 instead of 5 changes the answer noticeably. With 500 values, dividing by 499 instead of 500 makes almost no difference.

A quick reference for every symbol that shows up in standard deviation formulas:

Once you can read the symbols, the formulas read like a recipe rather than a wall of math.

Variance has its own pair of symbols, and they hint at the relationship between variance and standard deviation:

• σ² is population variance.
• s² is sample variance.

The squared notation is a hint: variance is standard deviation squared, and standard deviation is the square root of variance. Once you have one, you have the other. σ = √(σ²) and s = √(s²).

Take the small data set used across the other guides:

Map the symbols to the actual numbers:

• x is any one of the values (2, 4, 4, 4, 5, 5, 7, or 9).
• n = 8 (or N = 8 if these are the whole population).
• μ or x̄ = 5 (the mean: 40 / 8).
• Σ(x − μ)² = 32 (the sum of squared deviations from the mean).

Plug those into the formulas:

• σ = √(32 / 8) = √4 = 2
• s = √(32 / 7) ≈ √4.57 ≈ 2.14

Same data, two answers, depending on which formula and which symbol you reach for. For a step-by-step walkthrough of every step in detail, see Standard Deviation Formula. For a longer plain-language take on what the result actually means, see What Is Standard Deviation?

• σ is population standard deviation. s is sample standard deviation.
• μ is the population mean. x̄ is the sample mean.
• N is the population count. n is the sample count.
• n − 1 is the denominator in the sample formula (Bessel's correction).
• Σ means “sum of.”
• Variance reuses the same letters with squared notation: σ² and s².
• For a fast number that skips the symbols entirely, the standard deviation calculator takes a list of numbers and returns the answer.