The nearest-rank method

The percentile values for the ordered list {15, 20, 35, 40, 50}

One definition of percentile, often given in texts, is that the P-th percentile ${\displaystyle (0<P\leq 100)}$ of a list of N ordered values (sorted from least to greatest) is the smallest value in the list such that no more than P percent of the data is strictly less than the value and at least P percent of the data is less than or equal to that value. This is obtained by first calculating the ordinal rank and then taking the value from the ordered list that corresponds to that rank. The ordinal rank n is calculated using this formula

${\displaystyle n=\left\lceil {\frac {P}{100}}\times N\right\rceil .}$

• Using the nearest-rank method on lists with fewer than 100 distinct values can result in the same value being used for more than one percentile.
• A percentile calculated using the nearest-rank method will always be a member of the original ordered list.
• The 100th percentile is defined to be the largest value in the ordered list.

The linear interpolation between closest ranks method

An alternative to rounding used in many applications is to use linear interpolation between adjacent ranks.

All of the following variants have the following in common. Given the order statistics

${\displaystyle \{v_{i},i=1,2,\ldots ,N:v_{i+1}\geq v_{i},\forall i=1,2,\ldots ,N-1\},}$

we seek a linear interpolation function that passes through the points ${\displaystyle (v_{i},i)}$. This is simply accomplished by

${\displaystyle v(x)=v_{\lfloor x\rfloor }+(x{\bmod {1}})(v_{\lfloor x\rfloor +1}-v_{\lfloor x\rfloor }),\forall x\in [1,N]:v(i)=v_{i}{\text{, for }}i=1,2,\ldots ,N,}$

where ${\displaystyle \lfloor x\rfloor }$ uses the floor function to represent the integral part of positive x, whereas ${\displaystyle x{\bmod {1}}}$ uses the mod function to represent its fractional part (the remainder after division by 1). (Note that, though at the endpoint ${\displaystyle x=N}$, ${\displaystyle v_{\lfloor x\rfloor +1}}$ is undefined, it does not need to be because it is multiplied by ${\displaystyle x{\bmod {1}}=0}$.) As we can see, x is the continuous version of the subscript i, linearly interpolating v between adjacent nodes.

There are two ways in which the variant approaches differ. The first is in the linear relationship between the rank x, the percent rank ${\displaystyle P=100p}$, and a constant that is a function of the sample size N:

${\displaystyle x=f(p,N)=(N+c_{1})p+c_{2}.}$

There is the additional requirement that the midpoint of the range ${\displaystyle (1,N)}$, corresponding to the median, occur at ${\displaystyle p=0.5}$:

${\displaystyle {\begin{aligned}f(0.5,N)&={\frac {N+c_{1}}{2}}+c_{2}={\frac {N+1}{2}}\\\therefore 2c_{2}+c_{1}&=1\end{aligned}},}$

and our revised function now has just one degree of freedom, looking like this:

${\displaystyle x=f(p,N)=(N+1-2C)p+C.}$

The second way in which the variants differ is in the definition of the function near the margins of the ${\displaystyle [0,1]}$ range of p: ${\displaystyle f(p,N)}$ should produce, or be forced to produce, a result in the range ${\displaystyle [1,N]}$, which may mean the absence of a one-to-one correspondence in the wider region. One author has suggested a choice of ${\displaystyle C={\tfrac {1}{2}}(1+\xi )}$ where ξ is the shape of the Generalized extreme value distribution which is the extreme value limit of the sampled distribution.

First variant, C = 1/2

The result of using each of the three variants on the ordered list {15, 20, 35, 40, 50}

(Sources: Matlab "prctile" function,^[9][10])

${\displaystyle x=f(p)={\begin{cases}Np+{\frac {1}{2}},\forall p\in \left[p_{1},p_{N}\right],\\1,\forall p\in \left[0,p_{1}\right],\\N,\forall p\in \left[p_{N},1\right].\end{cases}}}$

where

${\displaystyle p_{i}={\frac {1}{N}}\left(i-{\frac {1}{2}}\right),i\in [1,N]\cap \mathbb {N} }$

${\displaystyle \therefore p_{1}={\frac {1}{2N}},p_{N}={\frac {2N-1}{2N}}.}$

Furthermore, let

${\displaystyle P_{i}=100p_{i}.}$

The inverse relationship is restricted to a narrower region:

${\displaystyle p={\frac {1}{N}}\left(x-{\frac {1}{2}}\right),x\in (1,N)\cap \mathbb {R} .}$

The weighted percentile method

In addition to the percentile function, there is also a weighted percentile, where the percentage in the total weight is counted instead of the total number. There is no standard function for a weighted percentile. One method extends the above approach in a natural way.

Suppose we have positive weights ${\displaystyle w_{1},w_{2},w_{3},\dots ,w_{N}}$ associated, respectively, with our N sorted sample values. Let

${\displaystyle S_{N}=\sum _{k=1}^{N}w_{k},}$

the sum of the weights. Then the formulas above are generalized by taking

${\displaystyle p_{n}={\frac {1}{S_{N}}}\left(S_{n}-{\frac {w_{n}}{2}}\right)}$ when ${\displaystyle C=1/2}$,

or

${\displaystyle p_{n}={\frac {S_{n}-Cw_{n}}{S_{N}+(1-2C)w_{n}}}}$ for general ${\displaystyle C}$,

and

${\displaystyle v=v_{k}+{\frac {P-p_{k}}{p_{k+1}-p_{k}}}(v_{k+1}-v_{k}).}$

The 50% weighted percentile is known as the weighted median.