Step by step
The question
Question: find the outliers in 100, 104, 106, 108, 110, 111, 113, 150 and 60.
An outlier is a number far away from the others.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Put them in order
First we put them in order, smallest first: 60, 100, 104, 106, 108, 110, 111, 113 and 150.
2 Find the quartiles
The median is the middle number.
9 numbers: 4 on each side of the middle one.
The median is 108.
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: 100 and 104.
Q1 is halfway between them: we add them and divide by 2.
100 plus 104 is 204.
Divided by 2 is 102.
The upper half has two in the middle: 111 and 113.
Q3 is halfway between them: we add them and divide by 2.
111 plus 113 is 224.
Divided by 2 is 112.
3 Find the IQR
The IQR, the interquartile range, is Q3 minus Q1: how wide the middle half is.
112 minus 102 is 10.
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 10 is 15.
A fence marks where far begins.
The low fence: Q1 minus 15.
102 minus 15 is 87.
The high fence: Q3 plus 15.
112 plus 15 is 127.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
60 and 150 are outside the fences.
So the outliers are 60 and 150.
6 Check
Let's check each one against its fence.
60 is less than 87.
150 is more than 127.
So it is right!
7 Pick the choice
Our answer is outliers: 60 and 150.
That is choice C.
8 Why not the others
A is none.
Wrong: 60 and 150 are outside the fences.
B is 150 only.
Wrong: 60 is below the lower fence too.
D is 60, 100 and 150.
Wrong: 100 is inside the fences.
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.
Then: check.
Then: pick the choice.
Last: why not the others.