What do the symbols in the standard deviation formula mean?
In the standard deviation formula, xᵢ is each data value, x̄ (x-bar) is the mean, n is the number of values, Σ means “add them all up”, and s is the answer: the sample standard deviation. Squared, s² is the variance. Below, every symbol is filled in with real numbers so you can see what it stands for.
s = √( Σ(xᵢ − x̄)² ÷ (n − 1) ) σ = √( Σ(xᵢ − μ)² ÷ N )
The left formula is for a sample, the right for a whole population. We will use the eight values 2, 4, 4, 4, 5, 5, 7, 9.
What is xᵢ in standard deviation?
xᵢ is “the i-th value”. The subscript i is a position number, so for our data x₁ = 2, x₂ = 4 and so on up to x₈ = 9. Writing xᵢ lets the formula say “do this to every value” without listing them. Sometimes it is written as plain x, meaning the same thing.
What is n (and N)?
n is how many values are in the sample: here n = 8. A capital N is used for the size of a whole population. The sample formula divides by n − 1 = 7 rather than n, to correct for measuring deviations from the sample's own mean; the sample vs population page explains why.
What does Σ mean?
Σ (capital sigma) means “add up everything that follows, for every value of i”. Σxᵢ is the sum of the data, 40. Σ(xᵢ − x̄)² is the sum of the squared deviations, 32, also called the sum of squares. Capital Σ is an instruction, not a result; lowercase σ is the population standard deviation. They are easy to confuse.
What are x̄ and μ?
x̄ (“x-bar”) is the sample mean: Σxᵢ ÷ n = 40 ÷ 8 = 5. μ (“mu”) is the population mean, used when your data are the entire population. Both are found the same way; the symbol only records which kind of data you have.
What does (xᵢ − x̄) mean?
It is the deviation: how far one value is from the mean. For x₁ = 2 it is 2 − 5 = −3; for x₈ = 9 it is +4. Squaring each one and adding gives the 32 above. The page on why the formula has a square root explains why the deviations are squared.
What do s and σ mean?
- s is the sample standard deviation: √(32 ÷ 7) = 2.138.
- σ (lowercase sigma) is the population standard deviation: √(32 ÷ 8) = 2.
On a calculator these appear as Sx and σx. The standard deviation symbol page covers those labels, SD and std dev, and how to type each symbol.
What is s squared in statistics?
s² is the sample variance, the value inside the square root: 32 ÷ 7 = 4.571. σ² is the population variance, 32 ÷ 8 = 4. The square of the standard deviation is always the variance, and the square root of the variance is always the standard deviation. See standard deviation vs variance for when to use each.
What does capital S or lowercase s mean?
Both usually mean the sample standard deviation. Lowercase s is the textbook convention; capital S appears in some books and in the Sx label on TI and Casio calculators. A subscript names the variable: sₓ for the x values, s_y for the y values in a two-variable problem.
All the symbols at a glance
| Symbol | Read as | Meaning | In the example |
|---|---|---|---|
| xᵢ | x sub i | Each data value | 2, 4, …, 9 |
| n | n | Sample size | 8 |
| N | capital N | Population size | — |
| Σ | capital sigma | Sum of what follows | Σxᵢ = 40 |
| x̄ | x-bar | Sample mean | 5 |
| μ | mu | Population mean | — |
| s, Sx | s | Sample standard deviation | 2.138 |
| σ, σx | sigma | Population standard deviation | 2 |
| s² | s squared | Sample variance | 4.571 |
| σ² | sigma squared | Population variance | 4 |
To check your own numbers, enter them in the standard deviation calculator, which shows x̄, every deviation, s and σ step by step.
Related calculators
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Standard deviation symbol
σ vs s, Sx vs σx, and how to type each symbol.
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Standard deviation formula
Both formulas, every variant, and copy-paste versions.
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How to calculate standard deviation
The steps by hand, in order.
Common questions
What is xᵢ in standard deviation?
xᵢ (x sub i) is one individual value in the data set. The i is a counter: x₁ is the first value, x₂ the second, up to xₙ. In Σ(xᵢ − x̄)², you take each value in turn, subtract the mean and square the result.
What is n in the standard deviation formula?
n is the number of values in a sample. The sample formula divides by n − 1; the population formula uses N, the number of values in the whole population, and divides by N.
What is s squared in statistics?
s² is the sample variance: the sum of squared deviations divided by n − 1. It is the standard deviation before the square root is taken, so s = √s². The population version is σ².
What does capital S mean in statistics?
Usually the sample standard deviation, the same as lowercase s; some textbooks and calculators (Sx on a TI-84) write it with a capital. Capital Σ, by contrast, means “add up” and is not a standard deviation.
What is the square of the standard deviation?
The variance: s² for a sample, σ² for a population. For the data on this page, s = 2.138 and s² = 4.571.
