Is this method of calculation for the first quartile correct?

Question

Given the numbers:

$[1,2,3,4,5]$

I want to find the first quartile, so I first find the median $ = 3.$ Now I can split the numbers in the this way:

$(1,2)$

$3$

$(4,5)$

Now to find the first quartile or $Q_1$ I must find the median between $(1,2)$. So that is equal to $(1+2)/2 = 1.5$.

So according to my calculation above $Q_1=1.5$.

If I am correct then why when I search online do I get formulas for the first quartile as such:

$Q_1 = \lceil{\frac{1}{4}(n+1)}\rceil th \ term$

using that formula here $Q_1$ would be $\frac{1}{4} (6) = \lceil1.5\rceil = 2$ ,which is position $2$ which is $2$.

Now obviously $2 \neq 1.5$. So am I wrong in my above calculation ?

Also this brings a curious phenomenon. Does $Q_1$, $Q_3$ never need to be average like the median ? The median in case of even terms needs to be averaged. But why not the other quartiles ?

What am I missing here?

Answer 1

In this case $n = 5$, so when you calculate the position, it would be $\frac{1}{4}(6) = 1.5$. This would be the average between the first and second position, which is also $1.5$. It is notable that for some sets, whether the median is included in each lower and upper half affects the selection of the quartiles.