Let's say I have a set of any 10 numbers and I've calculated some percentiles for them, e.g. 40th, 45th, 50th. Then I have an 11th number that falls within the minimum and maximum values of the original 10.
Example
x=(90,100,108,110,111,115,116,118,120,134)
the 40th, 45th, and 50th percentiles are 110.6, 111.2, and 113 respectively.
The 11th number is 112. Can we assume that all numbers are present between 111 and 115 such that 112 has a percentile that can be calculated/estimated without adding 112 to the set. Adding 112 to the set changes N and then also changes the 40th, 45th, and 50th percentiles.
In the 11 number set, x=(90,100,108,110,111, 112, 115,116,118,120,134) The 40th, 45th, and 50th percentiles, calculated in the same manner as above, are now 111, 111.5, 112.
Asked another way, can we determine what percentile 112 is where all other percentiles from the original set of 10 do not change? Or is this fundamentally not what percentiles are?