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.

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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 the average squared distance from the mean — the spread measured in squared units.

Common stuck point: 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.

Sense of Study hint: Ask: Am I averaging the squared distances from the mean (and not taking the square root)?

Worked Examples

Example 1

medium

Calculate the population variance for the data set:

\{2, 4, 4, 4, 5, 5, 7, 9\}

Answer

\sigma^2 = 4

First step

Find the mean:

\mu = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 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

\sigma^2 = \frac{\sum x_i^2}{n} - \mu^2

for

\{1, 2, 3, 4, 5\}

Example 4

medium

Compute the sample variance

s^2 = \frac{1}{n-1}\sum(x_i - \bar{x})^2

for

\{2, 4, 6, 8\}

Example 5

medium

For

X

taking values

1, 2, 3

each with probability

1/3

, compute

\operatorname{Var}(X) = E[X^2] - (E[X])^2

Example 6

hard

Fair die: let

X

be the value rolled. Compute

\operatorname{Var}(X)

Example 7

hard

For a Bernoulli random variable with success probability

p = 0.3

, find

\operatorname{Var}(X)

Example 8

challenge

Show that for two values

\{a,b\}

, the population variance equals

\frac{(b-a)^2}{4}

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

Example 3

easy

Find the variance of

\{2, 4, 6\}

(population).

Example 4

easy

If standard deviation is 5, what is the variance?

Example 5

easy

Variance of

\{5, 5, 5\}

Example 6

easy

The mean of

\{1, 3\}

is 2. Find the population variance.

Example 7

easy

If variance is 36, what is the standard deviation?

Example 8

easy

Data measured in meters has variance 9. What are the units of the variance?

Example 9

easy

Which is larger spread: variance 4 or variance 16?

Example 10

easy

Population variance divides the squared-deviation sum by what?

Example 11

medium

Find the population variance of

\{1, 2, 3, 4, 5\}

Example 12

medium

Find the sample variance of

\{2, 4, 6, 8\}

Example 13

medium

A set has population variance 10. Multiply every value by 3. New variance?

Example 14

medium

A set has variance 7. Add 5 to every value. New variance?

Example 15

medium

A fair coin:

X=1

for heads (p=0.5),

0

for tails. Find

\text{Var}(X)

Example 16

medium

Population variance of

\{0, 0, 0, 4\}

Example 17

medium

Variance equals

E[X^2]-(E[X])^2

. If

E[X^2]=20

and

E[X]=4

, find the variance.

Example 18

medium

Two independent variables:

\text{Var}(X)=3

\text{Var}(Y)=5

. Find

\text{Var}(X+Y)

Example 19

medium

Population variance of

\{4, 4, 10, 10\}

Example 20

challenge

A data set has values

\{3, 5, 5, x\}

, where the population variance is computed using deviations from the value

5

(i.e., treating

5

as the reference mean). If the resulting variance equals

5

, find the positive value of

x

Example 21

challenge

Show the variance of

\{a, -a\}

(population) equals

a^2

Example 22

challenge

A binomial

X

has

n=12

p=0.25

. Find

\text{Var}(X)

and explain via additivity.

Example 23

easy

Find the population variance of

\{4, 4, 4, 4\}

Example 24

easy

Find the population variance of

\{0, 2\}

Example 25

medium

Find the population variance of

\{3, 6, 9\}

Example 26

medium

Each value of a data set is multiplied by

3

. If the original variance was

4

, what is the new variance?

Example 27

medium

Each value of a data set is increased by

10

. If the original variance was

25

, what is the new variance?

Example 28

medium

Find the population variance of

\{10, 10, 10, 14\}

Example 29

medium

A data set has population variance

\sigma^2 = 16

measured in seconds

^2

. What is the standard deviation, in seconds?

Example 30

medium

Find the population variance of

\{-2, 0, 2\}

Example 31

medium

A class of

5

students scored

\{70, 80, 90, 80, 80\}

. Find the population variance.

Example 32

hard

A data set of

n=10

has

\sum x_i = 50

and

\sum x_i^2 = 300

. Find the population variance.

Example 33

hard

X

and

Y

are independent with

\operatorname{Var}(X) = 4

and

\operatorname{Var}(Y) = 9

. Find

\operatorname{Var}(X + Y)

Example 34

hard

X

and

Y

are independent with

\operatorname{Var}(X) = 4

and

\operatorname{Var}(Y) = 9

. Find

\operatorname{Var}(X - Y)

Example 35

hard

Given

\operatorname{Var}(X) = 5

, find

\operatorname{Var}(2X + 7)

Example 36

hard

A sample variance is

s^2 = 12

for

n = 9

. What is

\sum (x_i - \bar{x})^2

Example 37

hard

The variance of

\{a, a+d, a+2d\}

(population) is

\frac{2d^2}{3}

. Verify for

a=1

d=2

Example 38

challenge

A data set of

n

values has mean

\mu

and variance

\sigma^2

. A new value equal to

\mu

is added. What is the new population variance?

Example 39

challenge

Three numbers have mean

5

and population variance

6

. Two of them are

2

and

5

. Find the third.

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Related Concepts

Mean Standard Deviation Standard Deviation

Background Knowledge

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

meanstandard deviation