Outliers (Deep) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Outliers (Deep).

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

Concept Recap

An outlier is a data value that lies unusually far from most other values, potentially indicating measurement error, a rare event, or an important exception.

The weird one that doesn't fit. Is it a mistake, or something interesting?

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: An outlier sits beyond $1.5\times\text{IQR}$ past the quartiles — unusually far from the rest of the data.

Common stuck point: The procedure for outliers (deep) is the easy part; the trap is eyeballing outliers without the fence. Asking "Does this value fall beyond $Q_1-1.5\times\text{IQR}$ or $Q_3+1.5\times\text{IQR}$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this value fall beyond $Q_1-1.5\times\text{IQR}$ or $Q_3+1.5\times\text{IQR}$ ?

Worked Examples

Example 1

medium

Data:

\{12, 15, 14, 13, 16, 14, 15, 85\}

. Use the

1.5 \times IQR

rule to determine if 85 is an outlier, and discuss whether it should be removed.

Answer

85 is an outlier (exceeds fence of 18.5). Investigate cause before removing.

First step

Sort data:

\{12, 13, 14, 14, 15, 15, 16, 85\}

;

n=8

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Example 2

hard

Calculate the effect of an outlier (value 200) on the mean and median for

\{10, 12, 11, 13, 12, 200\}

, comparing to the data without the outlier

\{10, 12, 11, 13, 12\}

Example 3

medium

A data set has mean

50

and standard deviation

4

. A value of

66

is observed. Compute its

z

-score and decide whether it's an outlier by the

|z| > 3

rule.

Example 4

medium

Which is more resistant to an outlier — the standard deviation or the IQR? Justify with a one-line example.

Example 5

medium

Why does a boxplot make outliers visually obvious?

Example 6

hard

A single value at

z = 5

is found in a sample of

1000

. Should it be deleted automatically?

Example 7

hard

Compare the IQR rule to the

|z| > 3

rule. When does the z-score rule fail?

Example 8

medium

Why might removing an outlier change a correlation from

0.30

0.85

Example 9

challenge

A factory measures bolt diameters (mm) with target

5.00

and SD

0.02

. A bolt reads

5.10

. The supervisor says 'just one bolt' and ignores it. Why is this a poor decision?

Example 10

medium

Find all outliers in

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

using the IQR rule.

Practice Problems

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

Example 1

easy

A data set has

Q_1 = 50

Q_3 = 70

. Determine if each value is an outlier: (a) 85, (b) 15, (c) 105.

Example 2

hard

A researcher finds that removing one outlier changes the correlation from 0.45 to 0.82. Discuss whether the outlier should be removed and what this dramatic change reveals.

Example 3

easy

3, 4, 5, 6, 40

, which value is the obvious outlier?

Example 4

easy

Does an outlier affect the mean or the median more?

Example 5

easy

With

Q_1=20

Q_3=40

, what is the IQR used in the

1.5\times

IQR outlier rule?

Example 6

easy

True or false: every outlier is a data-entry error that should be deleted.

Example 7

easy

Using fences

Q_1-1.5\,\text{IQR}

and

Q_3+1.5\,\text{IQR}

with

Q_1=10

Q_3=20

, IQR

=10

, what is the upper fence?

Example 8

easy

A z-score tells how many standard deviations a value is from the mean. Is a value with

z=4

likely an outlier?

Example 9

easy

Which spread measure is most distorted by a single extreme value: range or IQR?

Example 10

easy

A class scores mostly 70–90, but one student scored 5 by leaving the test blank. Is this outlier likely an error or a real value?

Example 11

medium

Data:

4, 7, 8, 9, 10, 11, 12, 50

. With

Q_1=7.5

and

Q_3=11.5

, use the

1.5\times

IQR rule to identify outliers.

Example 12

medium

Without the value

100

, data

10,12,14,16

has mean

13

. With

100

added, the mean becomes

30.4

. By how much did the single outlier shift the mean?

Example 13

medium

A boxplot shows whiskers to

5

and

25

and a dot at

40

. Given

Q_3=22

and IQR

=8

, confirm whether

40

is correctly flagged as an outlier.

Example 14

medium

Mean

=50

, SD

=5

. A value of

68

is observed. Compute its z-score and judge whether it is an outlier by the

|z|>3

rule.

Example 15

medium

Removing a high outlier from a right-skewed dataset: will the mean decrease more than the median, less, or about the same?

Example 16

medium

A sensor logs

20, 21, 19, 20, 200, 21

. Should you investigate

200

before deciding it is a glitch, and what is the robust center estimate?

Example 17

medium

Why can the range alone fail to detect an outlier that the

1.5\times

IQR rule catches?

Example 18

medium

Mean

=100

, SD

=10

. Two values are observed:

125

and

108

. Using

|z|>3

, which (if any) is an outlier?

Example 19

medium

In data

5,6,7,8,9

, you add a new value

50

. Does the IQR or the range change more, and why?

Example 20

challenge

Data:

1, 2, 2, 3, 3, 3, 4, 4, 5, 30

. Find

Q_1

Q_3

, the IQR, and use the

1.5\times

IQR rule to list all outliers.

Example 21

challenge

A dataset's mean is

52

with an outlier and

48

without it; the outlier value is

200

. How many data points (including the outlier) are there?

Example 22

challenge

Two analysts disagree: one says a

z=2.5

point is an outlier, the other says it is not. Using the common

|z|>3

threshold, who is right, and what does this reveal about outlier rules?

Example 23

easy

Find the upper outlier fence for a data set with

Q_1 = 30

and

Q_3 = 50

Example 24

easy

Find the lower outlier fence for a data set with

Q_1 = 18

and

Q_3 = 26

Example 25

medium

For the data

\{4, 5, 6, 7, 8, 9, 30\}

, use the

1.5 \times \text{IQR}

rule to decide whether

30

is an outlier.

Example 26

medium

Data:

\{20, 22, 23, 24, 25, 26, 27, 28, 60\}

. Using the IQR rule, is

60

an outlier?

Example 27

medium

Compute the median and the mean of

\{2, 3, 4, 5, 6, 100\}

Example 28

medium

Data has

Q_1 = 12

Q_3 = 28

. Is the value

40

an outlier by the

1.5 \times \text{IQR}

rule?

Example 29

hard

A typo records a person's age as

250

instead of

25

. Why might this be a worse problem for the mean than for the median?

Example 30

hard

For

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

identify the IQR fences and any outliers.

Example 31

medium

The fences are

-1

and

11

. Which of

-3, 0, 5, 12

are outliers?

Example 32

hard

Adding the value

1000

to a 10-element data set whose mean is

50

and SD is

5

. Roughly, what happens to the SD?

Example 33

easy

True or false: the IQR rule uses only

Q_1

Q_3

, and the IQR.

Example 34

medium

A class of 25 students has

Q_1 = 70

and

Q_3 = 90

. Should a score of

40

be flagged?

Example 35

medium

Compute the upper fence for

Q_1 = 11

Q_3 = 21

. Is

35

an outlier?

Example 36

hard

A scientist replaces an outlier

90

with the mean of the rest (

45

). Why is this controversial?

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Related Concepts

Variability Interquartile Range Box Plot

Background Knowledge

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

variabilityinterquartile range