Variance Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Variance.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The variance is the average of the squared deviations from the mean: \sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2. It is the square of the standard deviation.

Another spread measure—variance = \text{SD}^2. Same idea, different scale.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Variance is in squared units (if data is in meters, variance is in meters^2).

Common stuck point: Take square root of variance to get SD (back to original units).

Sense of Study hint: Ask yourself: do I have the SD already? If so, just square it. If not, find each deviation from the mean, square them, and average.

Worked Examples

Example 1

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

Solution

  1. 1
    Find the mean: \mu = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5
  2. 2
    Calculate squared deviations: (2-5)^2=9,\ (4-5)^2=1,\ (4-5)^2=1,\ (4-5)^2=1,\ (5-5)^2=0,\ (5-5)^2=0,\ (7-5)^2=4,\ (9-5)^2=16
  3. 3
    Sum squared deviations: 9+1+1+1+0+0+4+16 = 32
  4. 4
    Divide by n: \sigma^2 = \frac{32}{8} = 4

Answer

\sigma^2 = 4
Variance measures how spread out values are from the mean. By squaring the deviations we ensure positive contributions and penalize larger deviations more heavily. Standard deviation would be \sqrt{4}=2.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Calculate the population variance for: \{1, 3, 5, 7, 9\}.

Example 2

medium
A data set has n=5 values with mean \mu = 10 and \sum x_i^2 = 530. Find the variance using the computational formula \sigma^2 = \frac{\sum x_i^2}{n} - \mu^2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

meanstandard deviation