Step by step
The question
Question: find the outliers in 12, 14, 15, 16, 18, 19, 20 and 22.
An outlier is a number far away from the others.
1 Put them in order
First we put them in order, smallest first: 12, 14, 15, 16, 18, 19, 20 and 22.
2 Find the quartiles
8 numbers: two in the middle, 16 and 18.
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.
16 plus 18 is 34.
Divided by 2 is 17.
The lower half has two in the middle: 14 and 15.
Q1 is halfway between them: we add them and divide by 2.
14 plus 15 is 29.
Divided by 2 is 14.5.
The upper half has two in the middle: 19 and 20.
Q3 is halfway between them: we add them and divide by 2.
19 plus 20 is 39.
Divided by 2 is 19.5.
3 Find the IQR
The IQR is Q3 minus Q1: how wide the middle half is.
19.5 minus 14.5 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.
Careful!
We step out from the ends of the box, Q1 and Q3.
Not from the median.
The low fence: Q1 minus 7.5.
14.5 minus 7.5 is 7.
The high fence: Q3 plus 7.5.
19.5 plus 7.5 is 27.
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: 12 is above 7, and 22 is below 27.
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.