Step by step
The question
Question: find the outliers in 0, 10, 11, 12, 13, 14, 15, 16 and 30.
An outlier is a number far away from the others.
1 Put them in order
First we put them in order, smallest first: 0, 10, 11, 12, 13, 14, 15, 16 and 30.
2 Find the quartiles
9 numbers: 4 on each side of the middle one.
The median is 13.
The median cuts the list in two halves, 4 numbers each.
The median itself is in neither half.
Careful!
The median is not counted in the halves.
Each half has 4 numbers.
The lower half has two in the middle: 10 and 11.
Q1 is halfway between them: we add them and divide by 2.
10 plus 11 is 21.
Divided by 2 is 10.5.
The upper half has two in the middle: 15 and 16.
Q3 is halfway between them: we add them and divide by 2.
15 plus 16 is 31.
Divided by 2 is 15.5.
3 Find the IQR
The IQR is Q3 minus Q1: how wide the middle half is.
15.5 minus 10.5 is 5.
4 Find the fences
The rule: an outlier is more than 1.5 IQRs outside the box.
1.5 times 5 is 7.5.
The low fence: Q1 minus 7.5.
10.5 minus 7.5 is 3.
The high fence: Q3 plus 7.5.
15.5 plus 7.5 is 23.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
0 and 30 are outside the fences.
So the outliers are 0 and 30.
6 Check
Let's check each one against its fence.
0 is less than 3.
30 is more than 23.
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.