Outliers and boxplots

Warm-up

Three quick questions from earlier lessons.

Discover

$Q_1 = 20$ , $Q_3 = 30$ , so IQR $= 10$ . Is a value of 48 unusually large?

Yes, it is above $30 + 1.5 \times 10 = 45$
No: only values above twice $Q_3$ , 60, are outliers

Key idea

A rule of thumb for outliers, and a picture of the five-number summary.

1.5 × IQR rule: a value is an outlier if it is below

Q_1 - 1.5\,\text{IQR}

or above

Q_3 + 1.5\,\text{IQR}

. A boxplot draws the box from

Q_1

to

Q_3

, a line at the median, whiskers to the most extreme non-outliers, and outliers as separate points.

Worked example

$Q_1 = 14$ , $Q_3 = 26$ . The fences.

IQR

= 12

26 − 14.

Fences

14 - 18 = -4

and

26 + 18 = 44

1.5 × 12 = 18.

Outliers: below

-4

or above

44

Any value beyond them.

Your turn, guided

Your turn: $Q_1 = 40$ , $Q_3 = 50$ . The lower fence.

25

40 − 15.

Practice ★

A data set has $Q_1 = 24$ and $Q_3 = 32$ . Above which value is a data point an outlier by the $1.5 \times \text{IQR}$ rule?
A data set has $Q_1 = 36$ and $Q_3 = 66$ . Above which value is a data point an outlier by the $1.5 \times \text{IQR}$ rule?
A data set has $Q_1 = 19$ and $Q_3 = 57$ . Above which value is a data point an outlier by the $1.5 \times \text{IQR}$ rule?

Watch out!

Practice ★★

$Q_1 = 34$ and $Q_3 = 44$ . Below which value is a data point an outlier?
$Q_1 = 40$ and $Q_3 = 48$ . Below which value is a data point an outlier?

A boxplot of 81 values shows which summary?
1. The five-number summary (min, Q1, median, Q3, max), with outliers plotted separately
2. The mean and the standard deviation
3. How many values fall in each interval

Exit ticket

Three questions to check you have got it.

A data set has $Q_1 = 24$ and $Q_3 = 32$ . Above which value is a data point an outlier by the $1.5 \times \text{IQR}$ rule?
A data set has $Q_1 = 36$ and $Q_3 = 66$ . Above which value is a data point an outlier by the $1.5 \times \text{IQR}$ rule?
A data set has $Q_1 = 19$ and $Q_3 = 57$ . Above which value is a data point an outlier by the $1.5 \times \text{IQR}$ rule?

Summary card

Outliers and boxplots

Fences: $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$ .
Boxplot: five-number summary plus outliers.

Watch out: 1.5 × IQR, not 1 × IQR.

About this lesson