10% quantile vs. 90% quantile

Question

Is it correct to say that, the 10% quantile in any situation has values less than 90% quantile?

Thanks in advance for taking your time

Answer 1

Yes, unless there are very many tied observation in your sample.

Suppose you have data

x
76 77 77 78 78 79 80 81 81 81

Then the 10% quantile (10th percentile, first decile) is found in R statistical software as follows:

quantile(x, .1)
  10% 
 76.9 

And for the 90% quantile:

quantile(x,.9)
 90% 
 81 

The idea is to find values that divide the sorted sample into 10 'chunks' of approximately equal size. In a sample of size $n = 10,$ each observation might be its own 'chunk', so it is relatively easy to find divisions points. In my example, there are ties in the vicinity of the 90% quantile that complicate the process, so the software chooses 81. Different textbooks and software packages have slightly different rules for such compromises, but the differences are relatively unimportant for large samples.

Here is an example with nine observations so many ties that the 10% and 90% quantiles are difficult to decide upon:

y
3 50 50 50 50 50 50 50 73

quantile(y, .10)
 10% 
40.6 


quantile(y, .90)
 90% 
54.6  

The algorithm R uses for quantiles attempts to make a slight distinction. However, some textbooks and software might say the the 10% and 90% quantiles are both 50.

And if all ten observations are 50, there is no way to make a distinction.