Step by step
The question
Question: find the outliers in 18, 22, 25, 19, 30, 21, 24, 20 and 46.
An outlier is a number far away from the others.
1 Put them in order
First we put them in order, smallest first: 18, 19, 20, 21, 22, 24, 25, 30 and 46.
2 Find the quartiles
The median is the middle number.
9 numbers: 4 on each side of the middle one.
The median is 22.
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 quartiles cut the list into four quarters.
Q1 is the middle of the lower half.
Q3 is the middle of the upper half.
The lower half has two in the middle: 19 and 20.
Q1 is halfway between them: we add them and divide by 2.
19 plus 20 is 39.
Divided by 2 is 19.5.
The upper half has two in the middle: 25 and 30.
Q3 is halfway between them: we add them and divide by 2.
25 plus 30 is 55.
Divided by 2 is 27.5.
3 Find the IQR
The IQR, the interquartile range, is Q3 minus Q1: how wide the middle half is.
27.5 minus 19.5 is 8.
4 Find the fences
How far is far?
We need a rule.
The usual rule: more than 1.5 IQRs below Q1, or above Q3.
Everyone uses 1.5, so everyone finds the same outliers.
1.5 times 8 is 12.
A fence marks where far begins.
The low fence: Q1 minus 12.
19.5 minus 12 is 7.5.
The high fence: Q3 plus 12.
27.5 plus 12 is 39.5.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
46 is outside the fences.
So the outlier is 46.
6 Check
Let's check each one against its fence.
46 is more than 39.5.
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.