Step by step
The question
Question: find the outliers in 2, 15, 17, 18, 20, 21, 23 and 45.
An outlier is a number far away from the others.
1 Put them in order
First we put them in order, smallest first: 2, 15, 17, 18, 20, 21, 23 and 45.
2 Find the quartiles
8 numbers: two in the middle, 18 and 20.
The median goes between them.
It cuts the list in two halves, 4 numbers each.
The median is halfway between them: we add them and divide by 2.
18 plus 20 is 38.
Divided by 2 is 19.
The lower half has two in the middle: 15 and 17.
Q1 is halfway between them: we add them and divide by 2.
15 plus 17 is 32.
Divided by 2 is 16.
The upper half has two in the middle: 21 and 23.
Q3 is halfway between them: we add them and divide by 2.
21 plus 23 is 44.
Divided by 2 is 22.
3 Find the IQR
The IQR is Q3 minus Q1: how wide the middle half is.
22 minus 16 is 6.
4 Find the fences
The rule: an outlier is more than 1.5 IQRs outside the box.
1.5 times 6 is 9.
Careful!
We step out from the ends of the box, Q1 and Q3.
Not from the median.
The low fence: Q1 minus 9.
16 minus 9 is 7.
The high fence: Q3 plus 9.
22 plus 9 is 31.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
2 and 45 are outside the fences.
So the outliers are 2 and 45.
6 Check
Let's check each one against its fence.
2 is less than 7.
45 is more than 31.
So it is right!
Let's look back.
First: put them in order.
Then: find the quartiles.
Then: find the IQR.
Then: find the fences.
Then: find the outliers.
Last: check.