Outlier Detection Examples in Statistics

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

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

Concept Recap

Outlier detection is the process of identifying data points that are unusually far from the rest of the dataset, using techniques like the IQR rule, z-scores, or visual inspection of box plots and scatter plots. These anomalous values may indicate measurement errors, data entry mistakes, or genuinely extreme observations.

Outliers are data points that don't fit the pattern. A 7-foot student in a class of average heights, or a \$10 million house in a neighborhood of \$300k homes. They may be errors or genuinely unusual.

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: Outlier Detection asks how a value or feature behaves inside the full distribution.

Common stuck point: Students often know a procedure related to outlier detection but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Mean vs Median Mistakes

Worked Examples

Example 1

medium

Data

2, 4, 5, 6, 8, 9, 25

has

Q_1=4, Q_3=9

. Is

25

an outlier by the IQR rule?

Is 25 an outlier by the IQR rule? (Q₁=4, Q₃=9)

Answer

\text{Yes}

First step

IQR = 9 - 4 = 5

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard

A normally distributed quality test has mean

100

, SD

10

. How likely is a true measurement above

130

, under the

|z|>3

rule?

Example 3

easy

The data set is: 10, 12, 11, 13, 12, 14, 11, 50. Identify the outlier and explain how you know.

Example 4

medium

Test scores: 72, 75, 78, 80, 82, 85, 88, 90, 92, 95. A new student's score of 25 is added. How does this outlier affect the mean and median?

Practice Problems

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

Example 1

easy

By the IQR rule, an outlier lies below

Q_1 - 1.5\cdot IQR

or above which bound?

Example 2

easy

A data set has

Q_1 = 10

and

Q_3 = 20

. Find the IQR.

Find the IQR from this box plot.

Example 3

easy

With

Q_1=10

Q_3=20

IQR=10

, find the lower fence

Q_1 - 1.5\cdot IQR

Find the lower fence Q₁ − 1.5·IQR.

Example 4

easy

With

Q_1=10

Q_3=20

IQR=10

, find the upper fence

Q_3 + 1.5\cdot IQR

Find the upper fence Q₃ + 1.5·IQR.

Example 5

easy

Fences are

-5

(lower) and

35

(upper). Is the value

40

an outlier?

Fences are −5 and 35. Is 40 an outlier?

Example 6

easy

Using z-scores, a common rule flags a value as an outlier when

|z|

exceeds what threshold?

Example 7

easy

Should you always delete an outlier as soon as you find one?

Example 8

easy

A box plot shows an isolated point far beyond the right whisker. What does it represent?

Example 9

medium

Data:

3, 5, 7, 8, 9, 12, 50

has

Q_1=5

Q_3=12

. Is

50

an outlier by the IQR rule?

Is 50 an outlier? (Q₁=5, Q₃=12)

Example 10

medium

Data with

Q_1=20

Q_3=40

. What are both IQR-rule fences?

Find both IQR-rule fences for Q₁=20, Q₃=40.

Example 11

medium

A value is

8

, the mean is

20

, and the SD is

4

. Is it an outlier by the

|z|>3

rule?

Example 12

medium

Data:

1, 2, 2, 3, 3, 3, 4, 100

. Which value is the obvious outlier and why?

Example 13

medium

An outlier of

100

is removed from a small data set. What happens to the mean and the SD?

Example 14

medium

Why might using ONLY the z-score method miss outliers in a skewed data set?

Example 15

medium

Data:

5, 6, 7, 8, 9

with

Q_1=6

Q_3=8

. Is the maximum value

9

an outlier?

Is the maximum value 9 an outlier? (Q₁=6, Q₃=8)

Example 16

medium

A scatter plot point sits far from an otherwise tight linear trend. What is it called and what might it indicate?

Example 17

medium

Data

4, 6, 8, 10, 12

has

Q_1=5

Q_3=11

. Are there any outliers by the IQR rule?

Are there any IQR-rule outliers? (Q₁=5, Q₃=11)

Example 18

challenge

Data:

2, 4, 6, 8, 10, 12, 14, 16

with

Q_1=5

Q_3=13

. Find both fences and state whether any value is an outlier.

Example 19

challenge

For data

10, 12, 14, 16, 18, 20, 100

with

Q_1=12

Q_3=20

, show

100

is an outlier and explain why the mean is a poor center here.

Is 100 an outlier? (Q₁=12, Q₃=20)

Example 20

challenge

Explain why the IQR rule is more robust to extreme values than the z-score rule (one concise reason).

Example 21

easy

A data set has

Q_1 = 30

and

Q_3 = 50

. Find the IQR.

Find the IQR. (Q₁=30, Q₃=50)

Example 22

easy

With

Q_1=30, Q_3=50, IQR=20

, what is the upper fence?

Find the upper fence. (Q₁=30, Q₃=50, IQR=20)

Example 23

easy

With

Q_1=30, Q_3=50, IQR=20

, what is the lower fence?

Example 24

easy

The mean is

50

, SD is

5

. Compute the z-score for

x = 70

Example 25

easy

True or false: outliers should always be removed before analysis.

Example 26

easy

Data:

4, 5, 6, 7, 8, 9, 100

. Which value is most likely an outlier?

Example 27

medium

A value has z-score

-2.5

in a roughly normal distribution. By the

|z|>3

rule, is it an outlier?

Example 28

medium

Data:

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 50\}

. Find

Q_1, Q_3

, and check if

50

is an outlier.

Find Q₁, Q₃, and check if 50 is an outlier.

Example 29

medium

In a strongly right-skewed distribution, IQR-rule outliers tend to appear on which side more often?

Example 30

medium

For

Q_1 = 12, Q_3 = 20

, is

30

an outlier by the IQR rule?

Example 31

medium

A z-score is calculated as

z = -3.4

for some value

x

. What does that mean?

Example 32

medium

A box plot shows the median near the BOX bottom, a short lower whisker, and one point far above the upper whisker. Most likely shape and outliers?

What shape does this box plot suggest, and what is the isolated point?

Example 33

hard

Data

\{2, 4, 4, 5, 6, 7, 8, 9, 12\}

has

Q_1 = 4, Q_3 = 8

. Identify any IQR-rule outliers.

Identify any IQR-rule outliers. (Q₁=4, Q₃=8)

Example 34

hard

A teacher records

30

students' typing speeds. A z-score

|z| > 3

rule flags two values. What should the teacher do?

Example 35

hard

Why does the IQR rule generally identify FEWER outliers than the

|z|>3

rule for skewed data?

Example 36

hard

With

Q_1=25, Q_3=75

, find both IQR-rule fences.

Example 37

hard

A modified IQR rule uses fences at

Q_1 - 3\cdot IQR

and

Q_3 + 3\cdot IQR

. Compared to the standard

1.5\cdot IQR

rule, what does it flag?

Example 38

medium

For ages of attendees at a children's birthday party, the value

42

appears next to a long list of values from

4

10

. Is

42

likely an outlier?

Example 39

hard

For data

\{0.5, 1, 1.2, 1.5, 1.8, 2, 2.2, 5\}

with

Q_1 = 1.1, Q_3 = 2.1

, is

5

an outlier?

Is 5 an outlier? (Q₁=1.1, Q₃=2.1)

Example 40

challenge

A data set of

50

values is symmetric and approximately normal with

\mu = 0, \sigma = 1

. Roughly how many values would you expect to be flagged by the

|z|>3

rule?

Example 41

medium

A scientist records reaction times (ms): 245, 260, 255, 270, 250, 980, 265, 258. Use the

1.5 \times IQR

rule to determine if 980 is an outlier. Should it be removed from the analysis?

Example 42

hard

A data set has mean

\bar{x} = 100

and standard deviation

s = 15

. Using the z-score method, determine whether the values 60, 145, and 155 are outliers (using the threshold

|z| > 2

← Back to Outlier Detection Common Mistakes Practice Mode

Related Concepts

Interquartile Range (IQR)Z-Score (Standard Score)Mean vs Median

Background Knowledge

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

stat interquartile rangestat z score