Normal Distribution Examples in Math

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

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 normal distribution (also called the Gaussian distribution or bell curve) is a continuous probability distribution that is symmetric about its mean, with data tapering off equally on both sides following a precise mathematical rule.

The normal distribution describes data that clusters symmetrically around the mean with a characteristic bell shape — most values are near the mean, and extreme values become rapidly less likely.

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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: The normal distribution is the bell-shaped curve where values cluster at the mean and thin out evenly on both sides.

Common stuck point: The procedure for normal distribution is the easy part; the trap is applying the 68-95-99.7 rule to non-normal data. Asking "Is the data single-peaked, symmetric, and described by just a mean and a standard deviation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the data single-peaked, symmetric, and described by just a mean and a standard deviation?

Worked Examples

Example 1

medium

Scores on a test are normally distributed with mean

\mu = 75

and standard deviation

\sigma = 10

. What percentage of students scored between

65

and

85

Answer

\approx 68\%

First step

Identify the mean

\mu = 75

and standard deviation

\sigma = 10

. Check whether

65

and

85

are within one standard deviation.

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Example 2

medium

Heights of adult women are normally distributed with

\mu = 164

cm and

\sigma = 6

cm. What percentage of women are taller than

176

cm?

Example 3

medium

Cholesterol levels in adults are approximately normal with

\mu = 200

mg/dL and

\sigma = 25

mg/dL. What percent of adults have cholesterol between

150

and

250

mg/dL?

Example 4

medium

Test scores are normal with

\mu = 80

\sigma = 6

. A teacher wants to find the cutoff above which only the top

16\%

of students score. Find it.

Example 5

hard

A normal distribution has

\mu = 100

and

\sigma = 12

. Using the empirical rule, estimate

P(76 < X < 112)

Example 6

hard

Adult male heights are normal:

\mu = 70

in,

\sigma = 3

in. In a sample of

1000

men, about how many are taller than

76

inches?

Example 7

challenge

For two independent normal random variables

X \sim N(\mu_1, \sigma_1^2)

and

Y \sim N(\mu_2, \sigma_2^2)

, what is the distribution of

X + Y

Example 8

challenge

Explain why incomes are typically NOT modeled as normal, and give one alternative distribution often used.

Practice Problems

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

Example 1

hard

A machine fills bottles with a mean of

500

mL and standard deviation

4

mL (normally distributed). What percentage of bottles contain between

492

mL and

508

mL?

Example 2

medium

A standardized exam has scores that are approximately normal with mean

500

and standard deviation

100

. About what percent of students score between

400

and

700

Example 3

easy

In a normal distribution, what percent of data lies within 1 standard deviation of the mean?

Example 4

easy

About what percent of normal data lies within 2 standard deviations of the mean?

Example 5

easy

About what percent of normal data lies within 3 standard deviations of the mean?

Example 6

easy

A normal distribution is symmetric about which value?

Example 7

easy

For the standard normal distribution, what are the mean and standard deviation?

Example 8

easy

Heights are normal with mean

170

cm. What percent of people are taller than

170

cm?

Example 9

easy

Is income across a population typically well-modeled by a normal distribution?

Example 10

easy

Scores are normal with mean

100

, SD

15

. Between which values do about 68% of scores fall?

Example 11

medium

IQ scores are normal with mean

100

, SD

15

. What percent of people have IQ above

130

Example 12

medium

Heights are normal with mean

170

cm, SD

10

cm. What percent are between

160

and

180

cm?

Example 13

medium

Test scores are normal with mean

500

, SD

100

. What percent score below

400

Example 14

medium

Scores are normal with mean

70

, SD

5

. What percent score between

70

and

80

Example 15

medium

A normal distribution has mean

50

, SD

4

. About what percent lies above

58

Example 16

medium

Weights are normal, mean

60

kg, SD

8

kg. Between what two weights do about 95% fall?

Example 17

medium

In a normal distribution, what percent of data lies between

-1

and

+2

standard deviations from the mean?

Example 18

medium

A standard normal value of

z=1

corresponds to what cumulative percent below it (using the empirical rule)?

Example 19

medium

Scores are normal, mean

75

, SD

5

. About what percent score above

85

Example 20

challenge

Scores are normal, mean

200

, SD

20

. About what percent score between

160

and

240

, and what does this say about

160

and

240

Example 21

challenge

Two normal distributions share mean

0

but have SDs

1

and

3

. Which has more data within the interval

[-1,1]

, and why?

Example 22

challenge

A normal distribution has mean

\mu

and SD

\sigma

. A value lies at the 97.5th percentile. How many SDs above the mean is it?

Example 23

easy

For a standard normal distribution, what is the value of

\mu+\sigma

Example 24

easy

A normal distribution has

\mu = 40

and

\sigma = 5

. What percent of data lies between

35

and

45

Example 25

easy

Reaction times are normal with

\mu = 0.30

s and

\sigma = 0.05

s. About what percent are slower than

0.40

Example 26

easy

Birth weights are normal with mean

3.4

kg and SD

0.5

kg. Between what two weights do about

99.7\%

of babies fall?

Example 27

easy

Salaries are normal with

\mu = \$60{,}000

and

\sigma = \$8{,}000

. What percent earn less than $60{,}000?

Example 28

medium

SAT scores are normal with

\mu = 1050

and

\sigma = 200

. What percent score between

850

and

1450

Example 29

medium

Battery lifetimes are normal with mean

500

hours and SD

40

hours. What percent last more than

580

hours?

Example 30

medium

A normal distribution has

\mu = 12

and

\sigma = 2

. What value sits at the

84

th percentile?

Example 31

medium

A factory part has length normal with

\mu = 20

cm,

\sigma = 0.1

cm. What percent of parts are shorter than

19.8

cm?

Example 32

medium

IQ is normal with

\mu=100

\sigma=15

. What fraction of people have IQ between

85

and

130

Example 33

medium

Times to complete a marathon are normal with

\mu = 4

30

min,

\sigma = 30

min. What percent finish between

3

30

min and

5

30

min?

Example 34

hard

A factory's bolt diameters are normal with

\mu = 10

mm and

\sigma = 0.05

mm. Bolts outside

[9.9, 10.1]

mm are rejected. About what percent are rejected?

Example 35

hard

X

is normal with

\mu = 25

\sigma = 4

. Find the value

c

such that

P(X < c) = 0.025

Example 36

challenge

X

is normal with

\mu=50

\sigma=10

. Define

Y = 3X + 5

. What are the mean and SD of

Y

Example 37

challenge

A standard normal

Z

satisfies

P(Z > a) = 0.025

. Find

a

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Related Concepts

Mean Standard Deviation Z-Score Central Limit Theorem

Background Knowledge

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

meanstandard deviation