Descriptive Statistics
The Standard Deviation Symbol (Sigma): Signs & Notation
Learn the standard deviation symbol: σ for population, s for sample. Understand every sign for standard deviation and which symbol identifies each.
The standard deviation symbol depends on whether you are describing a whole population or a sample drawn from it. Use the Greek lowercase letter sigma (σ) for a population standard deviation and the Latin lowercase letter s for a sample standard deviation. Getting the symbol right tells a reader immediately which kind of standard deviation you are reporting, and it matters every time you read a textbook, interpret a research paper, or use a statistics calculator.
Whenever you meet the symbol standard deviation notation in print — σ or s — that single character also encodes which formula produced the number, so choosing the wrong one misrepresents the math you actually did.
This article walks through every sign for standard deviation you will encounter — σ, s, σ², s², and a few variants — explains what each one means, shows the matching formula, and works through a complete example with real numbers.
Why the Symbol Matters
Symbols are a shorthand that carries precise meaning. When a published study writes σ = 12.4, it asserts that 12.4 is the spread of an entire population, computed by dividing by N (the total population count). When it writes s = 12.4, it asserts the same number comes from a sample, computed by dividing by n − 1 (one less than the sample size, the so-called Bessel’s correction).
Swap the symbols and you misrepresent the math. Confuse them in software and you may pick the wrong formula. Recognising the difference between population and sample standard deviation signs is one of the first notation skills a statistics student develops.
The Two Core Symbols for Standard Deviation
σ — The Population Standard Deviation Symbol
σ (lowercase sigma, the eighteenth letter of the Greek alphabet) is the symbol for standard deviation when the data covers an entire population. You see it in:
- textbooks on descriptive statistics and probability theory
- manufacturing quality-control specifications (process capability indices Cp and Cpk use σ directly)
- the normal-distribution notation N(μ, σ²), where μ is the population mean and σ² is the population variance
- the empirical rule (“68% of values fall within 1σ of the mean”)
The population standard deviation formula using this symbol is:
σ = √( Σ(xᵢ − μ)² / N )
where N is the number of values in the population and μ (mu) is the population mean.
s — The Sample Standard Deviation Symbol
s (an ordinary Latin lowercase letter) is the symbol for standard deviation when the data is a sample — a subset chosen from a larger population. The sample formula uses n − 1 in the denominator rather than N:
s = √( Σ(xᵢ − x̄)² / (n − 1) )
where n is the sample size and x̄ (x-bar) is the sample mean.
The n − 1 denominator (Bessel’s correction) adjusts for the fact that a sample tends to underestimate the spread of the full population. Dividing by n − 1 instead of n gives an unbiased estimator of the true population variance.
You will see s used in:
- hypothesis tests (t-tests, F-tests)
- confidence intervals
- almost every real-world dataset, because you rarely have access to the entire population
Quick Reference Table
| Symbol | Name | Describes | Denominator |
|---|---|---|---|
| σ | Lowercase sigma | Population standard deviation | N |
| s | Latin s | Sample standard deviation | n − 1 |
| σ² | Sigma squared | Population variance | N |
| s² | s squared | Sample variance | n − 1 |
| μ | Mu | Population mean | — |
| x̄ | X-bar | Sample mean | — |
Variance Symbols: σ² and s²
Variance is simply the standard deviation squared, so the symbols follow the same pattern.
σ² (sigma squared) is the symbol of the population variance. s² is the symbol for sample variance. Many tests and formulas work with the squared form because it is mathematically easier to manipulate algebraically (variances add across independent random variables; standard deviations do not).
When a textbook writes σ² = 4, it means the population variance is 4, and therefore σ = 2. When it writes s² = 4.57, the sample standard deviation is s ≈ 2.14.
Additional Symbols for Standard Deviation You May Encounter
The pair σ / s covers the vast majority of statistics work, but a few specialised notations appear in specific contexts.
SD or StdDev — used in some spreadsheet headers and older APA-style results tables as plain-English abbreviations. They are not formal mathematical symbols; they always refer to a sample standard deviation in that context unless the author specifies otherwise.
σ̂ (sigma-hat) — an estimator of the population standard deviation calculated from a sample. The hat (^) denotes estimation. In practice this is often numerically equal to s, though in advanced work the distinction between an estimator and the statistic s can matter.
σₓ̄ (sigma sub x-bar) — the standard error of the mean, equal to σ / √n. It describes the spread of sample means, not individual data points. This is a different quantity from σ; the subscript distinguishes them.
RSD — relative standard deviation, also called the coefficient of variation (CV). Expressed as a percentage: RSD = (s / x̄) × 100 %. Used to compare variability across datasets with different units or scales.
Recognising these additional standard deviation signs prevents confusion when you move between introductory statistics and more advanced or applied work.
Which Symbol Identifies the Population Standard Deviation?
To answer this directly: σ (lowercase sigma) identifies the population standard deviation. This is the answer to the question “which of the following symbols identifies the population standard deviation?” that appears frequently in statistics courses and multiple-choice exams.
The rule is consistent across every major statistics textbook and curriculum:
- σ → population standard deviation (whole group, divide by N)
- s → sample standard deviation (subset, divide by n − 1)
- σ² → population variance
- s² → sample variance
If you see σ on a test question, it always refers to the population. If you see s, it always refers to the sample. No exceptions in standard notation.
Fully Worked Example: Computing Both σ and s
Using the dataset 2, 4, 4, 4, 5, 5, 7, 9 — a classic reference set in introductory statistics — the following steps show exactly how each formula and its symbol plays out in practice.
Step 1: Find the Mean
Sum all values: 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40
There are n = 8 values.
Mean = 40 / 8 = 5
For a population this is written μ = 5. For a sample, x̄ = 5.
Step 2: Subtract the Mean and Square Each Deviation
| Value (xᵢ) | Deviation (xᵢ − 5) | Squared deviation |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | +2 | 4 |
| 9 | +4 | 16 |
| Sum | 32 |
Step 3a: Population Standard Deviation (σ)
Divide the sum of squared deviations by N = 8:
σ² = 32 / 8 = 4
Take the square root:
σ = √4 = 2
So the population standard deviation symbol σ carries the value 2 for this dataset. The population variance symbol σ² carries the value 4.
Step 3b: Sample Standard Deviation (s)
Divide the same sum of squared deviations by n − 1 = 7 (Bessel’s correction):
s² = 32 / 7 ≈ 4.5714
Take the square root:
s = √4.5714 ≈ 2.1381
So the sample standard deviation symbol s carries the value approximately 2.1381 for this dataset. The sample variance symbol s² carries approximately 4.5714.
What the Numbers Mean
The values σ = 2 and s ≈ 2.14 are close but not identical. The difference exists because the sample formula’s n − 1 denominator inflates the result slightly — by design. A purely mechanical division by 8 would underestimate how spread out the whole population really is; dividing by 7 corrects for that bias.
In a research paper you would write: “For the sample (n = 8), s = 2.14.” You would write: “If these eight values constitute the entire population, σ = 2.00.”
Try the Calculator
Enter the dataset 2, 4, 4, 4, 5, 5, 7, 9 below and confirm you get the same results. The calculator reports both the sample and population standard deviations, matching the σ and s values above.
Calculator
Standard Deviation Calculator
For a wider display with more options, open the full standard deviation calculator. You can also browse all tools on the calculators hub.
How Standard Deviation Symbols Appear in Formulas
Understanding the symbol for standard deviation in isolation is useful, but you also need to recognise how σ and s appear inside larger statistical formulas.
Z-Score Formula
z = (x − μ) / σ
Here σ is the population standard deviation used to standardise an individual observation. If you only have sample data, replace σ with s and μ with x̄.
Standard Error of the Mean
σ(x̄) = σ / √n
The subscript x̄ distinguishes this from the plain σ. The standard error shrinks as sample size grows — the more data you collect, the more precise your estimate of the mean becomes.
Confidence Interval for a Population Mean (known σ)
x̄ ± z* · (σ / √n)
When σ is unknown (the typical case), statisticians substitute s and use the t-distribution instead of z. The symbol swap signals a real change in the math: the t-distribution has heavier tails to account for the additional uncertainty introduced by estimating the spread from the sample itself.
Normal Distribution Notation
X ~ N(μ, σ²)
This compact notation says that the random variable X follows a normal distribution with mean μ and variance σ². The use of σ² rather than σ is deliberate: many theoretical results about the normal distribution are stated in terms of variance. Convert to standard deviation by taking the square root.
Reading Standard Deviation Symbols in Different Contexts
Textbooks and Lectures
Introductory textbooks almost always present the population formula first (using σ), then the sample formula (using s). The underlying data in class exercises is treated as a population more often than in real research, so do not assume the textbook’s approach matches what you will do in practice.
Research Papers
In applied research, almost every dataset is a sample. Expect to see s (or SD in APA style) unless the author explicitly says they measured every member of a defined population. The American Psychological Association’s Publication Manual uses M for the mean and SD for standard deviation in results text, but the mathematical formulas behind those abbreviations are still s and x̄.
Spreadsheets and Software
Microsoft Excel and Google Sheets distinguish the two formulas through function names rather than Greek letters:
STDEV.S— uses n − 1 (the s formula, sample)STDEV.P— uses N (the σ formula, population)
Python’s statistics.stdev() uses n − 1 (sample). NumPy’s np.std() defaults to N (population) unless you pass ddof=1 to get the sample version. Knowing the symbol–formula correspondence helps you pick the right function.
Multiple-Choice Exam Questions
Statistics exams — from introductory courses through AP Statistics and into college courses — routinely test symbol identification. A typical question lists σ, s, μ, and x̄ and asks you to match each to its definition. The answer key is always: σ = population standard deviation, s = sample standard deviation.
Common Mistakes with Standard Deviation Symbols
Mixing Up σ and s
The most common error: using σ when you mean s, or vice versa. This is not just cosmetic. It tells a reader the wrong denominator was used. Always check: did the data cover the whole population (σ) or a subset (s)?
Confusing σ with the Sum Operator Σ
The uppercase sigma Σ (capital Greek sigma) is the summation operator. It means “add everything that follows for each index value.” The lowercase σ is the standard deviation. They look similar, especially in handwriting, but they are completely different operations. In the formula below, both appear:
σ = √( Σ(xᵢ − μ)² / N )
The Σ in the numerator is the summation. The σ on the left is the standard deviation. Context and case always distinguish them.
Treating SD, StdDev, and s as Interchangeable
In practice they almost always refer to the same quantity (sample standard deviation). However, SD is an informal abbreviation, not an internationally standardised symbol. When precision matters — in a thesis, a technical report, or any document following a formal style guide — write s or σ and state which one you are using.
Forgetting the Squared Form
Variance (σ² or s²) and standard deviation (σ or s) are often confused. Variance is in squared units; standard deviation is in the original units. If your data is in kilograms, σ² is in kilograms squared and σ is in kilograms. Standard deviation is more interpretable for this reason, which is why it is reported more often — but variance is what you compute first in the formula.
Frequently Asked Questions
What is the standard deviation symbol?
There are two main symbols: σ (lowercase Greek sigma) for the population standard deviation and s (lowercase Latin s) for the sample standard deviation. In informal or spreadsheet contexts you may also see “SD” or “StdDev” as abbreviations, but σ and s are the standard mathematical symbols.
What is the sign for standard deviation in a formula?
In formal statistical notation the sign for standard deviation is σ for population data and s for sample data. Both appear as the subject on the left side of the formula (e.g., σ = √(Σ(xᵢ − μ)² / N)). The variance forms are σ² and s².
What is the symbol for standard deviation on a calculator or in software?
Physical calculators (such as the TI-84) label the population standard deviation σx and the sample standard deviation Sx. In Excel, the functions are STDEV.P (population) and STDEV.S (sample). In Python/NumPy, use ddof=0 for population (σ) and ddof=1 for sample (s).
Which of the following symbols identifies the population standard deviation?
σ (lowercase sigma) identifies the population standard deviation. It is always σ, never s. If a multiple-choice question offers σ, s, μ, and x̄, only σ is the population standard deviation — μ is the population mean, s is the sample standard deviation, and x̄ is the sample mean.
Are there other symbols for standard deviation?
Yes. You may encounter σ̂ (sigma-hat, an estimator), σₓ̄ (standard error of the mean), RSD (relative standard deviation as a percentage), and SD (informal abbreviation). All describe spread, but each has a specific technical meaning. The most important pair for any introductory course remains σ and s.
Why does sample standard deviation use s and not σ?
The convention distinguishes the two situations mathematically. Greek letters (μ, σ, σ²) are reserved for population parameters — fixed, true values that describe the whole group. Latin letters (x̄, s, s²) are used for sample statistics — estimates computed from a subset. This Greek-versus-Latin convention runs throughout statistics: once you know it, you can decode most notation at a glance.
Can σ and s ever be equal?
Yes, numerically they can be equal, though it is unusual. As the sample size n grows very large, the difference between dividing by n and n − 1 becomes negligible, so s approaches σ. In practice, for n = 8 the two values differ (σ = 2.00 vs s ≈ 2.14 in the worked example above); for n = 1000 they would be nearly identical.
Summary
The standard deviation symbol system reduces to one core rule: Greek for population, Latin for sample. Regardless of how a question phrases it — the symbol of standard deviation, the sign for standard deviation, or which symbol identifies the population standard deviation — the answer always follows this rule. Use σ (and σ²) for a population, s (and s²) for a sample, and read every other symbol you meet (σ̂, σₓ̄, SD, RSD) as an estimator, a derived quantity, or an informal abbreviation of those two.
Carry this rule and the worked example through any statistics course and you will never confuse the standard deviation signs again. The numbers are straightforward arithmetic; the symbols are just a consistent language that makes results unambiguous across textbooks, papers, and software alike.
For authoritative reference, the NIST/SEMATECH e-Handbook of Statistical Methods provides detailed coverage of standard notation for location and spread statistics, including the population and sample standard deviation symbols used throughout this article: https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm.