Empirical Rule Examples: 43 Problems with Answers

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

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

Concept Recap

The empirical rule (also called the 68-95-99.7 rule) states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

Most data clusters near the center of a bell curve; the further from the mean, the rarer the value.

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: For a roughly normal distribution, the empirical rule gives the percentage of data inside one, two, and three standard deviations of the mean: about 68%, 95%, and 99.7%.

Common stuck point: Students often know a procedure related to empirical rule but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?

Worked Examples

Example 1

medium

Heights of adult men are normal with

\mu = 70

in,

\sigma = 3

in. Estimate the proportion of men taller than

76

in.

Answer

\approx 2.5 %

First step

76 = 70 + 2 \cdot 3 = \mu + 2\sigma

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Example 2

medium

In a normal distribution, what percent lies between

\mu - \sigma

and

\mu + 2\sigma

Example 3

hard

A normal distribution has

\mu = 500

\sigma = 100

. About what percent lies between

400

and

700

Example 4

hard

A factory makes bolts with normal length,

\mu = 50

mm,

\sigma = 0.5

mm. Bolts shorter than

49

mm are rejected. About what fraction is rejected?

Example 5

hard

Heights of women are normal with

\mu = 64

in,

\sigma = 3

in. About how many women out of

1000

are between

58

and

70

inches tall?

Example 6

challenge

A factory produces tablets whose weights are normal with

\mu = 500

mg and

\sigma = 8

mg. Tablets weighing less than

484

mg or more than

516

mg are rejected. Approximately what fraction of tablets pass inspection?

Example 7

hard

IQ scores are normal,

\mu = 100

\sigma = 15

. About what percent of people have IQ between

70

and

145

Example 8

challenge

For a standard normal distribution, estimate the percent of data between

-1.5

and

+1.5

standard deviations using linear interpolation between the

1

-SD and

2

-SD empirical landmarks, then compare to the true value.

Practice Problems

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

Example 1

easy

State the three percentages of the empirical rule.

Example 2

easy

About what percent of normal data lies within

2

standard deviations of the mean?

What percent lies within ±2σ?

Example 3

easy

About what percent of normal data lies within

3

standard deviations of the mean?

What percent lies within ±3σ?

Example 4

easy

Does the empirical rule apply to a heavily skewed distribution?

Example 5

easy

In a normal distribution, what percent of data lies OUTSIDE

2

standard deviations?

What percent lies outside ±2σ?

Example 6

easy

Are the empirical-rule percentages exact or approximate?

Example 7

easy

In a normal distribution, what percent lies BELOW the mean?

What percent lies below the mean?

Example 8

easy

What percent of normal data lies between the mean and

+1

standard deviation?

What percent lies between μ and μ+σ?

Example 9

medium

IQ scores are normal with

\mu=100

\sigma=15

. What percent of people have IQ between

85

and

115

IQ: μ

Example 10

medium

IQ is normal,

\mu=100

\sigma=15

. What percent of people have IQ above

130

IQ: μ

Example 11

medium

A normal distribution has

\mu=200

\sigma=20

. What percent of data lies between

160

and

240

Example 12

medium

Heights are normal,

\mu=170

\sigma=10

. What percent of people are between

150

and

190

cm?

Heights: μ

Example 13

medium

A normal distribution,

\mu=50

\sigma=10

. What percent lies between

50

and

70

Example 14

medium

A normal distribution,

\mu=80

\sigma=6

. What percent of data lies below

74

Example 15

medium

Why does the empirical rule fail for the time between bus arrivals (often right-skewed)?

Example 16

medium

A normal distribution,

\mu=100

\sigma=10

. What percent lies between

90

and

120

Example 17

medium

A normal distribution has

\mu=70

\sigma=5

. What percent of data lies above

80

Example 18

challenge

Scores are normal with

\mu=500

\sigma=100

. Estimate the percent scoring between

300

and

600

Example 19

challenge

In a normal distribution, what percent of data lies between

+1\sigma

and

+2\sigma

above the mean?

What percent lies between +1σ and +2σ?

Example 20

challenge

A factory's part lengths are normal with

\mu=50

mm,

\sigma=2

mm. Parts outside

46

54

mm are scrapped. What percent is scrapped?

Example 21

easy

For a normal distribution, what percent of data lies between

\mu - \sigma

and

\mu + \sigma

What percent lies within ±1σ?

Example 22

easy

A normal distribution has

\mu = 50

\sigma = 5

. About what percent of values fall between

45

and

55

Example 23

easy

In a normal distribution, what percent lies outside

\pm 3

SD?

What percent lies outside ±3σ?

Example 24

easy

For a normal distribution, what percent of data lies below

\mu - 2\sigma

What percent lies below μ−2σ?

Example 25

easy

In a normal distribution with

\mu = 0

\sigma = 1

, what percent lies between

-3

and

3

Standard normal: what percent lies between −3 and 3?

Example 26

medium

SAT scores are normal with

\mu = 1000

\sigma = 200

. What percent of test-takers score between

800

and

1200

SAT: μ

Example 27

medium

SAT scores are normal with

\mu = 1000

\sigma = 200

. What percent score above

1400

SAT: μ

Example 28

medium

A test is normal with

\mu = 75

\sigma = 10

. What range covers about

95\%

of scores?

Example 29

medium

A normal distribution has

\mu = 100

\sigma = 10

. What percent of data lies between

90

and

110

Example 30

medium

Normal:

\mu = 60

\sigma = 4

. About what percent of values lie below

52

Example 31

hard

In a normal distribution with

\mu = 50

\sigma = 4

, what percent of values lie between

42

and

58

Example 32

hard

Normal:

\mu = 200

\sigma = 25

. What percent of values lie above

250

Example 33

hard

In a normal distribution, what percent of values lie between

\mu + \sigma

and

\mu + 2\sigma

What percent lies between μ+σ and μ+2σ?

Example 34

hard

A normal distribution has

\mu = 0

\sigma = 1

. About what percent of values lie above

1

Standard normal: what percent lies above 1 (i.e., above μ+σ)?

Example 35

medium

A normal distribution has

\mu = 20

\sigma = 3

. What percent of data lies between

14

and

23

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Related Concepts

Normal Distribution

Background Knowledge

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

stat normal distribution