Teacher Guide: Identifying Outliers

Learn what an outlier is, how to find it with the 1.5 × IQR rule, and how it changes the mean, median and range.

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10 questions on Spread & Variability. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Explain what an outlier is in a data set
Calculate the lower and upper fences with the 1.5 × IQR rule and identify outliers
Show outliers correctly in a box plot
Describe how an outlier affects the mean, median, range and IQR
Decide whether an outlier is an error to correct or a real value to keep

Prerequisites

• Finding the median, Q1, Q3 and IQR (Range and Interquartile Range)
• Reading and drawing box plots (Box Plots)
• Calculating the mean of a data set
• Basic idea of standard deviation (Standard Deviation), for the z-score remark

Discussion Starters

1. A runner sets a new world record. Is that time an outlier? Should it be removed from the data?
2. Why do you think news reports often give the median house price instead of the mean?
3. If you remove an outlier from a data set, which changes more: the mean or the median? Why?
4. Can a value be an outlier in one data set and normal in another? Give an example.
5. Who should decide whether a strange value is a mistake, and what information would they need?

Common Misconceptions

Outliers are always mistakes and should be deleted

Remediation: Show real outliers (a record jump, a lottery winner's income) next to clear errors (a height of 1630 cm). Ask students to sort them and explain why only errors may be corrected or removed.

Only very large values can be outliers

Remediation: Use a data set with a low outlier, like quiz scores $3, 12, 13, \ldots$ . Have students compute both fences every time, not only the upper one.

The largest value in a data set is automatically an outlier

Remediation: Give a data set where the maximum is inside the upper fence (for example $95$ with fence $96.25$ ). Stress that the fences decide, not the position in the list.

Differentiation Ideas

For Struggling Students:

• Give Q1 and Q3 directly so students practise only the fences
• Use a fill-in template: IQR, 1.5 × IQR, lower fence, upper fence, outliers
• Show the fences on a number line so students see which dots lie outside

For On-Level Students:

• Find outliers from raw data sets of 8 to 12 values
• Draw box plots with outliers as separate dots
• Compare the mean and median with and without an outlier

For Advanced Students:

• Compare the 1.5 × IQR rule with the 2-standard-deviation rule on the same data
• Find a value that would be exactly on a fence and explain why it is not an outlier
• Research a real data set (city temperatures, sports results) and argue whether each outlier is an error or real

Standards Alignment

6.SP.B.5c (CCSS.MATH.CONTENT.6.SP.B.5.C)
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern
6.SP.B.5d (CCSS.MATH.CONTENT.6.SP.B.5.D)
Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered
HSS-ID.A.3 (CCSS.MATH.CONTENT.HSS.ID.A.3)
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers)

Lesson Resources

visualInteractive Box Plot
Students add and remove values and watch outliers appear as separate dots
activityClass Data Hunt
Collect a class data set (for example, hand spans or travel times), find the fences and look for outliers
worksheetError or Real?
Short data sets with outliers; students find the fences and decide whether each outlier is an error or a real value

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

An outlier is a data value that lies unusually far from the rest of the data. It is much smaller or much larger than the other values.

The 1.5 × IQR rule

To decide in a fair way whether a value is an outlier, we build two fences from the quartiles:

\text{Lower fence} = Q_1 - 1.5 \times \text{IQR}

\text{Upper fence} = Q_3 + 1.5 \times \text{IQR}

where

\text{IQR} = Q_3 - Q_1

. Any value below the lower fence or above the upper fence is an outlier. A value exactly on a fence is not an outlier.

How to find outliers

1.Order the data and find $Q_1$ and $Q_3$ (medians of the lower and upper half, as in Range and Interquartile Range)
2.Calculate $\text{IQR} = Q_3 - Q_1$
3.Multiply: $1.5 \times \text{IQR}$
4.Lower fence $= Q_1 - 1.5 \times \text{IQR}$ , upper fence $= Q_3 + 1.5 \times \text{IQR}$
5.Every value outside the fences is an outlier

Outliers in a box plot

In a box plot, outliers are drawn as separate dots. The whiskers stop at the smallest and largest values that are not outliers.

Why outliers matter

An outlier pulls the mean and stretches the range, but it barely moves the median or the IQR. That is why the median and IQR are called resistant measures.

Other methods exist too: with the standard deviation, a value more than 2 or 3 standard deviations from the mean is often called unusual (see Standard Deviation).

Worked Examples

A data set has

Q_1 = 20

and

Q_3 = 30

. Which of the values

2

18

44

and

50

are outliers?

Find the IQR

\text{IQR} = Q_3 - Q_1 = 30 - 20

→

\text{IQR} = 10

Multiply the IQR by 1.5

1.5 \times 10

→

15

Find the lower fence

Q_1 - 15 = 20 - 15

→ Lower fence

= 5

Find the upper fence

Q_3 + 15 = 30 + 15

→ Upper fence

= 45

Compare each value with the fences

2 < 5

(outside),

18

and

44

are between

5

and

45

(inside),

50 > 45

(outside) → Outliers:

2

and

50

Answer: Only $2$ and $50$ are outliers. The values $18$ and $44$ lie inside the fences $5$ and $45$ .

Common Mistakes

Using $Q_3 + \text{IQR}$ instead of $Q_3 + 1.5 \times \text{IQR}$

Why it's wrong: Forgetting the factor $1.5$ makes the fences too close to the box, so ordinary values get marked as outliers.

Correct: Always multiply first: $1.5 \times \text{IQR}$ , then subtract it from $Q_1$ and add it to $Q_3$ .

Building the fences from the median, the minimum or the maximum

Why it's wrong: The rule measures distance from the box, and the box ends at $Q_1$ and $Q_3$ .

Correct: Lower fence $= Q_1 - 1.5 \times \text{IQR}$ , upper fence $= Q_3 + 1.5 \times \text{IQR}$ . Never start from the median or the extremes.

Drawing the whisker all the way to the outlier

Why it's wrong: Then the outlier looks like part of the normal spread, and the box plot hides it.

Correct: The whisker ends at the largest (or smallest) value that is not an outlier. Draw each outlier as a separate dot.

Deleting every outlier automatically

Why it's wrong: An outlier can be a real and important value, like a record time or a very large order.

Correct: Remove or correct a value only when it is an error (typo, broken instrument, wrong unit). Keep real outliers and report them, or use the median and IQR.

Why It Matters

Outliers show up wherever people collect data:

Banks: a card payment far above your usual purchases is flagged as possible fraud
Factories: a bottle filled with much less juice than the others is an outlier that quality control removes from the line
Science: a thermometer reading of $52°C$ on a mild day is probably a broken sensor, not a heat wave
Sports: a world record is a real outlier, and it is the most interesting value of all
Salaries: one very high salary makes the mean salary look much higher than what most people earn

Finding outliers helps you spot mistakes, notice something special, and choose the right measure (mean or median) to describe your data honestly.

Real World Applications

Spotting Card Fraud

Banks compare each new card payment with the customer's usual payments. A payment above the upper fence can trigger a security check.

Example:

A customer's payments have $Q_1 = 20$ dollars and $Q_3 = 60$ dollars. A payment of 450 dollars is far above the upper fence, so the bank sends a text message to confirm it.

1Try It Yourself

A customer's usual card payments have $Q_1 = 20$ dollars and $Q_3 = 60$ dollars. Suddenly a payment of 450 dollars appears.

What is the upper fence for this customer's payments (in dollars)?

Step 1: Write the mathematical expression

Calculate $Q_3 + 1.5 \times \text{IQR}$ :

Quality Control in a Factory

Juice bottles should all hold about the same amount. Bottles below the lower fence are taken off the line and checked.

Example:

If most bottles hold between 498 mL and 502 mL, a bottle with 480 mL is clearly underfilled and the machine may need repair.

2Try It Yourself

Fill volumes of juice bottles have $Q_1 = 498$ mL and $Q_3 = 502$ mL.

What is the lower fence (in mL)?

Step 1: Write the mathematical expression

Calculate $Q_1 - 1.5 \times \text{IQR}$ :

Reporting a Typical Salary

News reports often give the median salary instead of the mean, because a few very high salaries pull the mean up.

Example:

In a small company, five employees earn between 30 and 38 thousand dollars a year and the owner earns 160 thousand dollars. The owner's salary is an outlier.

3Try It Yourself

Yearly salaries in thousands of dollars: $30, 32, 34, 36, 38, 160$ .

What is the mean salary (in thousands of dollars)?

Step 1: Write the mathematical expression

Add all six salaries and divide by 6:

Key Takeaways

1An outlier is a value unusually far from the rest of the data, either very small or very large
21.5 × IQR rule: lower fence $= Q_1 - 1.5 \times \text{IQR}$ , upper fence $= Q_3 + 1.5 \times \text{IQR}$
3Values below the lower fence or above the upper fence are outliers; a value exactly on a fence is not
4In a box plot, outliers are separate dots and the whiskers stop at the last values inside the fences
5Outliers pull the mean and stretch the range, while the median and IQR barely change
6Correct or remove an outlier only if it is an error; keep real outliers and report them

Frequently Asked Questions

Why 1.5 and not 1 or 2?

The factor

1.5

was chosen by the statistician John Tukey because it works well in practice: it flags values that are really unusual without marking too many ordinary values. Some books also use

3 \times \text{IQR}

for "extreme" outliers.

Can a data set have no outliers, or more than one?

Yes to both. Many data sets have no outliers at all, and some have several, on one side or on both sides (like the temperatures

2

and

52

in the examples).

Is the minimum or maximum always an outlier?

No. The minimum and maximum are just the smallest and largest values. They are outliers only if they lie outside the fences. For example, in

60, 65, 70, 72, 75, 78, 80, 95

the upper fence is

96.25

, so the maximum

95

is not an outlier.

Is there another way to find outliers?

Yes. With the standard deviation you can measure how many standard deviations a value is from the mean (its z-score). Values more than 2 or 3 standard deviations away are often called unusual. The 1.5 × IQR rule is used for box plots because it does not depend on the mean, which outliers distort.

Glossary

Outlier: A data value that lies unusually far from the other values; by the 1.5 × IQR rule, a value below the lower fence or above the upper fence
Lower fence: The boundary $Q_1 - 1.5 \times \text{IQR}$ ; values below it are outliers
Upper fence: The boundary $Q_3 + 1.5 \times \text{IQR}$ ; values above it are outliers
1.5 × IQR rule: A method for finding outliers: build fences 1.5 IQRs below $Q_1$ and 1.5 IQRs above $Q_3$
Resistant measure: A statistic that changes very little when an outlier is added, such as the median and the IQR

Formula Card

Interquartile range

\text{IQR} = Q_3 - Q_1

The spread of the middle 50% of the data

Lower fence

Q_1 - 1.5 \times \text{IQR}

Values below the lower fence are outliers

Upper fence

Q_3 + 1.5 \times \text{IQR}

Values above the upper fence are outliers

Teacher Guide: Identifying Outliers

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

The 1.5 × IQR rule

How to find outliers

Outliers in a box plot

Why outliers matter

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Spotting Card Fraud

Quality Control in a Factory

Reporting a Typical Salary

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Interquartile range

Lower fence

Upper fence

More in This Topic

Range and Interquartile Range

Standard Deviation

Calculating Standard Deviation