I have a slight confusion with the current method of calculation of a quantile for a give ungrouped distribution. To give you an example, i shall refer to calculation of a Quartile, but this doubt applies to any quantile.
Let's take a distribution as follows:
0,2,4,9,10
n=5
The formulae to calculate the lower quartile and upper quartile are as follows:
Q1=(n+1)/4
Q3=3*(n+1)/4
So as per these formulae, the upper and lower quartile should appear at positions 1.5 and 4.5
We can all agree here that the median of the distribution is 4 (i.e., position 3)
However, to derive the upper and lower quartile values, we have to assume that the end point values of the range are 0 and 10 (i.e, position 1 and 5). Hence by this logic, and assuming the end point values of the range are 0 and 10 (position 1 and 5) and the median as 4 (position 3), we should get the upper and lower quartile positions as 2 (i.e. (1+3)/2) and 4 (i.e. (5+3)/2) respectively.
However, going by the accepted formula we see that we see that the lower quartile 1.5 is a median position between a non-existant position 0 and 3, and the upper quartile 4.5 is a median position between position 3 and a non-existant position 6.
Can anyone please clarify why the non-existant positions are assumed? Perhaps I'm looking at it from a wrong angle, so please let me know.