What Is Variance Symbol?
The variance symbol depends on what your data represent. A lowercase s² is the sample variance, an estimate built from a subset. A lowercase Greek σ² is the population variance, used when you have every value in the group.
Enter your values and the calculator returns both versions side by side, with the divisor each one uses: n − 1 for s² and n for σ². The symbols table below lists the related notation, including Var(X) and the standard deviations.
Pick the symbol that matches your situation and quote it with the number. Writing σ² for a sample is a common slip that readers notice.
How to Use the Variance Symbol Calculator
- Enter your data.
- Read s² and σ².
- Choose the symbol that matches your data.
- Check the symbols table.
Variance Symbol Formula
Sample variance
Population variance
Variance Symbol Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| s² | Sample variance, divides by n − 1 | |
| σ² | Population variance, divides by n | |
| x̄ | Sample mean | |
| μ | Population mean | |
| n | Number of values |
Variance Symbol Worked Example
For 6, 9, 7, 12, 10, 8, 11 the mean is 9 and SS = 9 + 0 + 4 + 9 + 1 + 1 + 4 = 28. So s² = 28 ÷ 6 = 4.667 and σ² = 28 ÷ 7 = 4.
With the example values loaded in the calculator above, the steps are:
- Mean x̄ = 9; sum of squared deviations SS = 28.
- Sample variance s² = SS ÷ (n − 1) = 28 ÷ 6 = 4.6666667.
- Population variance σ² = SS ÷ n = 28 ÷ 7 = 4.
Result: Sample variance s² = 4.6666667; Population variance σ² = 4.
How to Read the Two Variances
Both variances use the same sum of squares, SS = Σ(x − mean)². They differ only in the divisor: s² divides by n − 1 and σ² divides by n. For the data 6, 9, 7, 12, 10, 8, 11, SS = 28, so s² = 28 ÷ 6 = 4.667 and σ² = 28 ÷ 7 = 4.
The sample value is always larger. The ratio s² ÷ σ² equals n ÷ (n − 1), which is 7 ÷ 6 here. The gap matters for small samples and almost disappears for large ones.
Where Variance Symbol Is Used
Choosing the symbol
Decide whether your data are a sample or the whole population and use s² or σ² to match.
Reading a textbook
Compare the notation in your course with the table of variance symbols, including Var(X) and the standard deviations.
Checking software output
Match the two values to the functions in a spreadsheet, where VAR.S gives s² and VAR.P gives σ².
Report writing
Label the variance in a table so a reader knows which divisor you used.
Common Variance Symbol Mistakes to Avoid
- Use s² when the data are a sample from a larger group, which is the usual case.
- Use σ² only when you have every member of the population, such as all students in one class.
- A variance is in squared units. Report the standard deviation if you want the original units.
- Do not mix symbols. Writing σ² for a sample is a common slip.
Variance Symbol Terms and Definitions
- σ²
- The usual symbol for population variance.
Variance Symbol FAQs
What is the symbol for variance?
The sample variance is s² and the population variance is σ². In probability the variance of a random variable is written Var(X) or σ²_X.
What is the difference between s² and σ²?
s² is computed from a sample and divides the sum of squares by n − 1. σ² is for a whole population and divides by n. Dividing by n − 1 corrects the tendency of a sample to underestimate the spread.
What does Var(X) mean?
It is the variance of the random variable X: the expected squared distance of X from its mean, E[(X − μ)²]. It is the same quantity as σ² for that variable.
Why is variance squared?
Squaring makes every deviation positive so they cannot cancel, and it weights large deviations more. The price is squared units, which is why the standard deviation is often quoted instead.