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Here is my reasoning:

Let's assume the exclusive definition of a percentile here. Suppose we have a set of numbers {10,20,30,40} and we want to calculate value of its 90th percentile. But such value doesn't seem to exist. If we take a number from (30;40] interval, then it will be a 75th percentile because 3/4=0.75. We can't take a number that is >40. So value of 75th percentile is the closest one to value 90th percentile that we can get.

Now let's try the inclusive definition of a percentile on the same set of numbers in order to see if 90th will exist. In this case any number from interval [30;40) would be a 75th percentile. For the inclusive definition of a percentile 100th percentile DOES exist and we will get it if we choose 40, although we still can't choose any number >40. So 100th percentile is closest we will get to a 90th percentile in this case.

It shows that we can't get a 90th percentile in both cases. Does it mean that set of numbers {10,20,30,40} has NO 90th percenile? And consequently, that for some datasets certain percentiles don't exist?

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  • $\begingroup$ Yes that is right. $\endgroup$ – aman May 27 at 14:39
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It depends on the definition of the $n$th percentile. You could define it as the value for which exactly $n\%$ of the data is below it. For this dataset, any percentile other than 25, 50, 75, and 100 is undefined. You could also define it as the smallest value in the list greater than or equal to $n\%$ of the dataset. Then the 90th percentile of $\{10,20,30,40\}$ is $40$. It is also the 100th percentile and the 76th percentile.

But the real problem here is that percentiles aren't very meaningful for small datasets. Why would you want to talk about the 90th percentile of a 4-element dataset?

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Your argument is correct. That said, in everyday numeracy percentiles are used to make sense of large datasets, like nedian family income in the United States. Then a record or two error if the number of records isn't exactly divisible by $100$ does not matter.

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  • $\begingroup$ What do you mean "record isn't divisible by 100"? And what do we do in this case? $\endgroup$ – user161005 May 27 at 15:05
  • $\begingroup$ He means that if the number of items in the dataset is not divisible by $100$. Say you have a dataset of $101$ items. There is no one item that represents any given percentile-the $90^{th}$ largest item is larger than $\frac {89}{101}$ of the items, which is not a whole number, but we would call it the $89^{th}$ percentile anyway. There are a number of definitions of quantiles, which all agree in the main characteristics but differ in the details. If you care about the details you need to specify which one you are using. $\endgroup$ – Ross Millikan May 27 at 15:12
  • $\begingroup$ @RossMillikan And what are these definitions? I skimmed through the article that you linked, but didn't find them. $\endgroup$ – user161005 May 27 at 15:24
  • $\begingroup$ I remembered that they were in the article, but I don't see them now either. They have to do with what you do to break ties and what to do when there isn't a data item called out. In your original example, some would linearly interpolate between the data points, for example. Sorry, I don't know where the definitions are. $\endgroup$ – Ross Millikan May 27 at 15:26

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