Step by step
The question
Question: find the outliers in 20, 22, 23, 25, 26, 28, 29 and 31.
An outlier is a number far away from the others.
1 Put them in order
First we put them in order, smallest first: 20, 22, 23, 25, 26, 28, 29 and 31.
2 Find the quartiles
8 numbers: two in the middle, 25 and 26.
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.
25 plus 26 is 51.
Divided by 2 is 25.5.
The lower half has two in the middle: 22 and 23.
Q1 is halfway between them: we add them and divide by 2.
22 plus 23 is 45.
Divided by 2 is 22.5.
The upper half has two in the middle: 28 and 29.
Q3 is halfway between them: we add them and divide by 2.
28 plus 29 is 57.
Divided by 2 is 28.5.
3 Find the IQR
The IQR is Q3 minus Q1: how wide the middle half is.
28.5 minus 22.5 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.
22.5 minus 9 is 13.5.
The high fence: Q3 plus 9.
28.5 plus 9 is 37.5.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
Every dot is between the fences.
No number is outside the fences.
So there are no outliers.
6 Check
Let's check the ends: 20 is above 13.5, and 31 is below 37.5.
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.