Step by step
The question
Question: find the outliers in 1, 30, 32, 33, 35, 36, 38, 40 and 70.
An outlier is a number far away from the others.
1 Put them in order
First we put them in order, smallest first: 1, 30, 32, 33, 35, 36, 38, 40 and 70.
2 Find the quartiles
9 numbers: 4 on each side of the middle one.
The median is 35.
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: 30 and 32.
Q1 is halfway between them: we add them and divide by 2.
30 plus 32 is 62.
Divided by 2 is 31.
The upper half has two in the middle: 38 and 40.
Q3 is halfway between them: we add them and divide by 2.
38 plus 40 is 78.
Divided by 2 is 39.
3 Find the IQR
The IQR is Q3 minus Q1: how wide the middle half is.
39 minus 31 is 8.
4 Find the fences
The rule: an outlier is more than 1.5 IQRs outside the box.
1.5 times 8 is 12.
The low fence: Q1 minus 12.
31 minus 12 is 19.
The high fence: Q3 plus 12.
39 plus 12 is 51.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
1 and 70 are outside the fences.
So the outliers are 1 and 70.
6 Check
Let's check each one against its fence.
1 is less than 19.
70 is more than 51.
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.