Variance Formula

The variance is the average of the squared deviations from the mean: σ^2 = (1/n)Σ (x_i - x̄)^2.

The Formula

σ2=∑(x−μ)2n

When to use: Another spread measure—variance =SD2. Same idea, different scale.

Quick Example

Data: {2,4,6}. Mean =4. Deviations: −2,0,2. Squared: 4,0,4. Variance =(4+0+4)/3≈2.67.

Notation

σ2 for population variance, s2 for sample variance

What This Formula Means

The variance is the average of the squared deviations from the mean: σ2=1n∑(xi−xˉ)2. It is the square of the standard deviation.

Another spread measure—variance =SD2. Same idea, different scale.

Formal View

σ2=1n∑i=1n(xi−μ)2 (population); s2=1n−1∑i=1n(xi−xˉ)2 (sample)

Worked Examples

Example 1

medium
Calculate the population variance for the data set: {2,4,4,4,5,5,7,9}.

Answer

σ2=4

First step

1
Find the mean: μ=2+4+4+4+5+5+7+98=408=5

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Example 2

hard
Two investments have the same mean return of 8%. Investment A returns: {6,7,8,9,10}%. Investment B returns: {2,5,8,11,14}%. Calculate the variance of each and interpret.

Example 3

medium
Use the computational formula σ2=∑xi2n−μ2 for {1,2,3,4,5}.

Common Mistakes

  • Taking the square root and calling it variance — that result is the standard deviation; variance is before the root.
  • Averaging the deviations without squaring — raw deviations sum to zero, so square them first.
  • Mixing up the divisor — population variance divides by n, sample variance (s2) divides by n−1.

Why This Formula Matters

Variance is the algebra-friendly form of spread: variances of independent quantities add, which is why it underlies regression, ANOVA, and the central limit theorem. Standard deviation is what you report to humans; variance is what the formulas run on. Recognizing it by "Am I averaging the squared distances from the mean (and not taking the square root)?" — rather than by familiar numbers — is what lets a student tell it apart from standard deviation and mean and range in a mixed problem set.

Frequently Asked Questions

What is the Variance formula?

The variance is the average of the squared deviations from the mean: σ2=1n∑(xi−xˉ)2. It is the square of the standard deviation.

How do you use the Variance formula?

Another spread measure—variance =SD2. Same idea, different scale.

What do the symbols mean in the Variance formula?

σ2 for population variance, s2 for sample variance

Why is the Variance formula important in Math?

Variance is the algebra-friendly form of spread: variances of independent quantities add, which is why it underlies regression, ANOVA, and the central limit theorem. Standard deviation is what you report to humans; variance is what the formulas run on. Recognizing it by "Am I averaging the squared distances from the mean (and not taking the square root)?" — rather than by familiar numbers — is what lets a student tell it apart from standard deviation and mean and range in a mixed problem set.

What do students get wrong about Variance?

The procedure for variance is the easy part; the trap is taking the square root and calling it variance. Asking "Am I averaging the squared distances from the mean (and not taking the square root)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Variance formula?

Before studying the Variance formula, you should understand: mean, standard deviation.

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