Think about it for a short while, anyone can see that there is no definitive way to define what is a percentile. What is surprising is how many different ways we can define it.

In a 1996 paper, there is a generic model proposed for what a percentile function can be and nine different variations enumerated.

# Terms and definition

Percentile, or any quantile, is a rank statistic. Assume we have a set of measurements $X_1,\cdots,X_n$ and the version in ascending order of magnitude $\mathbf{X}=\{X_{(1)}, X_{(2)}, \cdots, X_{(n)}\}$. Roughly speaking, the quantile function $Q(p)$, defined over $p\in[0,1]$, gives some value $X_{(k)}\in\mathbf{X}$ such that the number of measurements is fraction $p$ of $|\mathbf{X}|$:

We call this a percentile function if $p$ is defined in percentage.

There are several characteristics that the quantile function $Q(p)$ may satisfy. As mentioned in Hyndman & Fan (1996),

1. Surjective from $[0,1]$ to $\mathbf{X}$: $Q(p)$ is continuous on $p$
2. Approximation of distribution function: $pn$ gives the lower bound of the number of observations $X_i$ in $\mathbf{X}$ less than $Q(p)$,
$|\{X_i: X_i\in\mathbf{X}, X_i\le Q(p)\}| \ge pn$
3. Quantile are based on samples arranged evenly and symmetrically on $[0,1]$:
$\begin{gather} |\{X_i: X_i\in\mathbf{X}, X_i\le Q(p)\}| = |\{X_i: X_i\in\mathbf{X}, X_i\ge Q(1-p)\}| \\ Q^{-1}(X_{(k)}) + Q^{-1}(X_{(n-k+1)}) = 1 \end{gather}$
4. Min and max in $\mathbf{X}$ are in $[0,1]$:
$Q^{-1}(X_{(1)}) \ge 0, Q^{-1}(X_{(n)}) \le 1$
5. $Q(0.5)$ is the median

# Classes of quantile functions

Quantile function can look like a step function, or a piecewise linear continuous function.

Hyndman & Fan proposed a generic form for quantile function:

such that

## Step functions

Wikipedia covers one step function variant, the nearest rank method, which defined $Q(p)=X_{(j)}$ such that $j=\lceil pn \rceil$. This is similar to the 1st definition of Hyndman & Fan, which has the properties that

• step function with jumps on $p=j/n$ for integral $j$.
• caglad: each step is a horizontal line segment with closed end on left and open end on right, i.e. $Q(j/n) = X_{(j+1)}$ instead of $X_{(j)}$ (Wikipedia article’s definition use $Q(j/n)=X_{(j)}$).

We can have variation on how to handle the end points of steps to this function. The 2nd definition of Hyndman & Fan is to use midpoint, i.e., the step function still have jumps on $p=j/n$ but has the particular cases defined $Q(j/n)=\frac{1}{2}(X_{(j+1)} + X_{(j)})$.

Formally, the 1st definition has $m=0$ (so jumps on $p=j/n$), $\gamma=1$ if $g>0$ or 0 otherwise. The 2nd definition has $m=0$, $\gamma=\frac{1}{2}$ for $g=0$ and $\gamma=1$ for $g>0$.

The 3rd definition of the paper gives an alternative way to jump: The step function has jumps at the midpoint of $p=j/n$ and $p=(j+1)/n$ instead. And each step is a line segment that is closed on both end if it is a step on $j/n$ for $j$ an even number (even steps) or the line segment is open on both end otherwise (odd steps). Formally, $m=-\frac{1}{2}$, $\gamma=0$ if $g=0$ and $j$ is even, or $\gamma=1$ otherwise. This is the implementation in SAS.

## Interpolated functions

This is not a step function but a monotonically increasing continuous function. Instead of jumps, the function is a connected line segments. Wikipedia gives the following generic form: $Q(p)=X_{(j)}$ such that

where $c\in[0,1]$ and for $j\notin\mathbb{N}$. $X_{(j)}$ is interpolated from adjacent measurements $X_{(\lfloor j \rfloor)}$ and $X_{(\lceil j \rceil)}$. Hence this is not a step function.

This is used in Excel. The function PERCENTILE.INC(array,k) has $c=1$ and $p=\frac{j-1}{n-1}$ and PERCENTILE.EXC(array,k) has $c=0$ and $p=\frac{j}{n+1}$. The former, also as the 7th definition in Hyndman & Fan, defined the 0th and 100th percentile as the min and max value of $\mathbb{X}$ respectively. The latter, also known as the 6th definition in Hyndman & Fan, however, has the min above 0th percentile and the max below 100th percentile (assuming $\mathbf{X}$ is a sample and the sample min/max is not the population min/max).

Matlab has $c=\frac{1}{2}$ or $p=\frac{j-1/2}{n}$. Same as PERCENTILE.EXC(array,k), the min and max are not on 0th and 100th percentile. This is also the 5th definition in Hyndman & Fan.

These three variation can be understood from a histogram setting: Assume we have to layout the $n$ data points as histogram on $[0,1]$:

1. The $c=1$ case: We can split $[0,1]$ into $n-1$ equal segments and put the $n$ data points at the $n$ segment boundaries (including 0 and 1)
2. The $c=\frac{1}{2}$ case: We can split $[0,1]$ into $n$ equal segments and put the $n$ data points at the middle of each segment
3. The $c=0$ case: We can split $[0,1]$ into $n+1$ equal segments and put the $n$ data points at the $n$ segment boundaries (not including 0 and 1)

After the data points are placed, $Q(p)$ is the function that connected them with straight line segments.

Hyndman & Fan has three more variation of the interpolation function:

• 4th definition: $p=\frac{j}{n}$, max is the 100th percentile. Histogram representation is to split $[0,1]$ into $n$ equal segments like the $c=\frac{1}{2}$ way above but locate the data point at right end of each segment
• 8th definition: $p=\frac{k-1/3}{n+1/3}$. Histogram representation is to split $[0,1]$ into $n+\frac{1}{3}$ equal segments, and from the $\frac{2}{3}$-th segment onward up to $(n-\frac{1}{3})$-th segment, there are $n-1$ segments apart. We place the $n$ data points evenly in between, including the ends on the $\frac{2}{3}$-th and $(n-\frac{1}{3})$-th. This is the optimal estimator for median, if $\mathbf{X}$ is a sample.
• 9th definition: similar to above but defines $p=\frac{k-3/8}{n+1/4}$. This is a better estimator for mean for samples drawn from normal distribution.

# Inverse quantile function

In excel, the inverse of percentile function is called percentile rank (PERCENTILERANK.INC(array,x) and PERCENTILERANK.EXC(array,x)). The inverse function is similarly confusing to define and additionally imposed another problem.

In case of duplicated values in the order statistics $\mathbf{X}$, e.g., $X_{(j)}=X_{(j+1)}$, what should be $Q^{-1}(X_{(j)})$?

Given the inverse property, we knows that $p_{\min} \le Q^{-1}(X_{(j)}) \le p_{\max}$ for $p_{\min} = \min(p: Q(p)=X_{(j)})$ and $p_{\max}$ defined similarly.

Excel uses $Q^{-1}(X_{(j)}) = p_{\min}$. But it is equally reasonable for $Q^{-1}(X_{(j)}) = p_{\max}$ and $Q^{-1}(X_{(j)}) = \frac{1}{2}(p_{\max}+p_{\min})$.

## Bibliographic data

@article{
title = "Sample Quantiles in Statistical Packages",
author = "Rob J. Hyndman and Yanan Fan",
year = "1996",
journal = "The American Statistician",
volume = "50",
number = "4",
month = "Nov",
pages = "361-365",
doi = "http://dx.doi.org/10.1080/00031305.1996.10473566",
}