Step by step
The question
Question: find the third quartile, Q3, of 12, 5, 8, 20, 15, 9, 11, 7 and 14.
There are four choices.
First we solve it, the normal way.
Then we pick.
Q3 is the 75th percentile.
About 75% of the numbers are below it.
1 Put them in order
In order, smallest to biggest.
Small green numbers count the positions: 1, 2, 3, and so on.
2 Find the position
Why the rule uses n plus 1: think of the middle number.
With 9 numbers, the middle one is the 5th: 4 on each side.
That is half of 9 plus 1.
So for any percent, we take that percent of n plus 1.
The position i: 75/100, times n plus 1.
n is how many numbers: 9.
75/100 is 0.75.
9 plus 1 is 10.
0.75 times 10 is 7.5.
Careful!
The rule uses n plus 1, not n.
That keeps the middle in the middle.
3 Count to the position
7.5 is not whole.
It is between the 7th and the 8th numbers: 14 and 15.
So we take the number halfway between them: add, then divide by 2.
14 plus 15 is 29.
Divided by 2: 14.5.
4 Check
Let's check: 7 of the 9 numbers are below 14.5.
That is about 78%, close to 75%.
So it makes sense!
5 Pick the choice
Our answer is Q3 = 14.5.
That is choice B.
6 Why not the others
A is 14.
Wrong: put the median into the upper half: leave it out.
That is the Careful mistake from before.
C is 11.
Wrong: that is the median.
D is 7.5.
Wrong: that is Q1.
Let's look back.
First: put them in order.
Then: find the position.
Then: count to the position.
Then: check.
Then: pick the choice.
Last: why not the others.