Step by step
The question
Question: find the outliers in 52, 55, 57, 58, 60, 61, 63, 64 and 90.
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: 52, 55, 57, 58, 60, 61, 63, 64 and 90.
2 Find the quartiles
The median is the middle number.
9 numbers: 4 on each side of the middle one.
The median is 60.
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: 55 and 57.
Q1 is halfway between them: we add them and divide by 2.
55 plus 57 is 112.
Divided by 2 is 56.
The upper half has two in the middle: 63 and 64.
Q3 is halfway between them: we add them and divide by 2.
63 plus 64 is 127.
Divided by 2 is 63.5.
3 Find the IQR
The IQR, the interquartile range, is Q3 minus Q1: how wide the middle half is.
63.5 minus 56 is 7.5.
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 7.5 is 11.25.
A fence marks where far begins.
The low fence: Q1 minus 11.25.
56 minus 11.25 is 44.75.
The high fence: Q3 plus 11.25.
63.5 plus 11.25 is 74.75.
5 Find the outliers
On a number line: the fences in red, and a dot for each number.
90 is outside the fences.
So the outlier is 90.
6 Check
Let's check each one against its fence.
90 is more than 74.75.
So it is right!
7 Pick the choice
Our answer is outlier: 90.
That is choice B.
8 Why not the others
A is none.
Wrong: 90 is above the upper fence.
C is 52 and 90.
Wrong: 52 is inside the fences.
D is 64 and 90.
Wrong: 64 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.