Step by step
The question
Question: find the first quartile, Q1, of 22, 15, 30, 18, 27, 35, 12, 25 and 20.
There are four choices.
First we solve it, the normal way.
Then we pick.
Q1 is the 25th percentile.
About 25% 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: 25/100, times n plus 1.
n is how many numbers: 9.
25/100 is 0.25.
9 plus 1 is 10.
0.25 times 10 is 2.5.
Careful!
The rule uses n plus 1, not n.
That keeps the middle in the middle.
3 Count to the position
2.5 is not whole.
It is between the 2nd and the 3rd numbers: 15 and 18.
So we take the number halfway between them: add, then divide by 2.
15 plus 18 is 33.
Divided by 2: 16.5.
4 Check
Let's check: 2 of the 9 numbers are below 16.5.
That is about 22%, close to 25%.
So it makes sense!
5 Pick the choice
Our answer is Q1 = 16.5.
That is choice C.
6 Why not the others
A is 18.
Wrong: put the median into the lower half: leave it out.
B is 22.
Wrong: that is the median.
D is 28.5.
Wrong: that is Q3.
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.