Mfcttrf

Are there 99 percentiles, or 100 percentiles? And are they groups of numbers, or divider lines, or pointers to individual numbers?

I suppose the same question would apply for quartiles or any quantile.

I have read that the index of a number at a particular percentile(p), given n items, is i = (p / 100) * n

That suggests to me that there are 100 percentiles.. because supposing you have 100 numbers(i=1 to i=100), then each would have an index(1 to 100).

If you had 200 numbers, there'd be 100 percentiles, but would each refer to a group of two numbers. Or 100 dividers excluding either the far left or far right divider 'cos otherwise you'd get 101 dividers. Or pointers to individual numbers so the first percentile would refer to the second number, (1/100)*200=2 And the hundredth percentile would refer to the 200th number (100/100)*200=200

I have sometimes heard of there being 99 percentiles though..

Google shows the oxford dictionary that says of percentile- "each of the 100 equal groups into which a population can be divided according to the distribution of values of a particular variable." and "each of the 99 intermediate values of a random variable which divide a frequency distribution into 100 such groups."

Wikipedia says "the 20th percentile is the value below which 20% of the observations may be found" But does it actually mean "the value below or equal to which, 20% of the observations may be found" i.e. "the value for which 20% of the values are <= to it". If it were just < and not <=, then By that reasoning, the 100th percentile would be the value below which 100% of the values may be found. I have heard that as an argument that there can be no 100th percentile, because you can't have a number where there are 100% of the numbers below it. But I think maybe that argument that you can't have a 100th percentile is incorrect and is based an error that the definition of a percentile involves <= not <. (or >= not >). So the hundredth percentile would be the final number and would be >= 100% of the numbers.

One nice way to treat this is to start with simple math and work backwards to the more complicated case of real data. Let's start with PDF's, CDF's, and inverse CDF's (also known as quantile functions). The $x$th quantile of a distribution with pdf $f$ and cdf $F$ is $F^-1(x)$. Suppose the $z$th percentile is $F^-1(z/100)$. For a uniform 0,1 distribution, the 100th and 0th percentiles are ill-defined since $F^-1$ is only unique when $F$ is not constant. For a normal distribution, they do not exist (or they "are" $pm infty$).

For continuous distributions, non-extreme quantiles exist and are unique. For a discrete distribution such as the Poisson distribution, most percentiles don't exist because for most $z/100$, there is no $y$ with $F(y) = z/100$.

When it comes to real data, all distributions are discrete. (The empirical CDF of runif(100) or np.random.random(100) has 100 increments clustered around 0.5.) We still want to have a useful concept of quantiles and percentiles. So, we can define a quantile as any consistent estimator of the corresponding theoretical quantile. For example, the median (the 50th percentile or 0.5 quantile) of the sample 3,4, 5, 6, 7, 8 can be any number between 5 and 6. If you draw 2n samples from a unif(3,8) distribution and take any number between the nth and (n+1)th sample, you will converge on 5.5 as n increases.

(In practice, I would always compute the median of 3, 4, 5, 6, 7, 8 as 5.5 because that particular median is also a trimmed mean, which means it has other good properties as an estimator.)

I was taught that an observation in the nth percentile was greater than n% of observations in the dataset under consideration. Which to me implies that there is no 0th or 100th percentile. No observation can be greater than 100% of observations because it forms part of that 100% (and a similar logic applies in the case of 0).

But I unfortunately have no source for this that I can point you to.