# Is this method of calculation for the first quartile correct?

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?

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.