Practice Variance in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

Use the computational formula

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

for

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

Example 2

medium

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

Example 3

medium

Find the sample variance of

\{2, 4, 6, 8\}

Example 4

hard

The variance of

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

(population) is

\frac{2d^2}{3}

. Verify for

a=1

d=2

Example 5

medium

A data set has population variance

\sigma^2 = 16

measured in seconds

^2

. What is the standard deviation, in seconds?

Example 6

medium

Find the population variance of

\{-2, 0, 2\}

Example 7

hard

Fair die: let

X

be the value rolled. Compute

\operatorname{Var}(X)

Example 8

challenge

A binomial

X

has

n=12

p=0.25

. Find

\text{Var}(X)

and explain via additivity.

Example 9

challenge

Show the variance of

\{a, -a\}

(population) equals

a^2

Example 10

medium

Compute the sample variance

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

for

\{2, 4, 6, 8\}

Example 11

hard

A sample variance is

s^2 = 12

for

n = 9

. What is

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

Example 12

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 13

medium

Find the population variance of

\{3, 6, 9\}

Example 14

easy

If variance is 36, what is the standard deviation?

Example 15

challenge

Three numbers have mean

5

and population variance

6

. Two of them are

2

and

5

. Find the third.

Example 16

easy

The mean of

\{1, 3\}

is 2. Find the population variance.

Example 17

medium

For

X

taking values

1, 2, 3

each with probability

1/3

, compute

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

Example 18

medium

Population vs. sample variance: which divides by

n-1

Example 19

hard

A data set of

n=10

has

\sum x_i = 50

and

\sum x_i^2 = 300

. Find the population variance.

Example 20

hard

X

and

Y

are independent with

\operatorname{Var}(X) = 4

and

\operatorname{Var}(Y) = 9

. Find

\operatorname{Var}(X - Y)

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Related Concepts

Mean Standard Deviation Standard Deviation