Percentile

Instatistics,ak-thpercentile,also known aspercentile scoreorcentile,is ascorebelow whicha givenpercentagekof scores in itsfrequency distributionfalls ( "exclusive"definition) or a scoreat or below whicha given percentage falls ( "inclusive"definition). Percentiles are expressed in the sameunit of measurementas the input scores,notinpercent;for example, if the scores refer tohuman weight,the corresponding percentiles will be expressed in kilograms or pounds. In thelimitof an infinitesample size,the percentile approximates thepercentile function,the inverse of thecumulative distribution function.

Percentiles are a type ofquantiles,obtained adopting a subdivision into 100 groups. The 25th percentile is also known as the firstquartile(Q₁), the 50th percentile as themedianor second quartile (Q₂), and the 75th percentile as the third quartile (Q₃). For example, the 50th percentile (median) is the scorebelow(orat or below,depending on the definition) which 50% of the scores in the distribution are found.

A related quantity is thepercentile rankof a score, expressed inpercent,which represents the fraction of scores in its distribution that are less than it, an exclusive definition. Percentile scores and percentile ranks are often used in the reporting oftest scoresfromnorm-referenced tests,but, as just noted, they are not the same. For percentile ranks, a score is given and a percentage is computed. Percentile ranks are exclusive: if the percentile rank for a specified score is 90%, then 90% of the scores were lower. In contrast, for percentiles a percentage is given and a corresponding score is determined, which can be either exclusive or inclusive. The score for a specified percentage (e.g., 90th) indicates a score below which (exclusive definition) or at or below which (inclusive definition) other scores in the distribution fall.

Definitions

There is no standard definition of percentile; however, all definitions yield similar results when the number of observations is very large and theprobability distributionis continuous.In the limit, as the sample size approaches infinity, the 100p^thpercentile (0<p<1) approximates the inverse of thecumulative distribution function(CDF) thus formed, evaluated atp,aspapproximates the CDF. This can be seen as a consequence of theGlivenko–Cantelli theorem.Some methods for calculating the percentiles are given below.

The normal distribution and percentiles

Representation of thethree-sigma rule.The dark blue zone represents observations within onestandard deviation(σ) to either side of themean(μ), which accounts for about 68.3% of the population. Two standard deviations from the mean (dark and medium blue) account for about 95.4%, and three standard deviations (dark, medium, and light blue) for about 99.7%.

The methods given in thecalculation methodssection (below) are approximations for use in small-sample statistics. In general terms, for very large populations following anormal distribution,percentiles may often be represented by reference to a normal curve plot. The normal distribution is plotted along an axis scaled tostandard deviations,or sigma ($\sigma$) units. Mathematically, the normal distribution extends to negativeinfinityon the left and positive infinity on the right. Note, however, that only a very small proportion of individuals in a population will fall outside the −3σto +3σrange. For example, with human heights very few people are above the +3σheight level.

Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, −3σis the 0.13th percentile, −2σthe 2.28th percentile, −1σthe 15.87th percentile, 0σthe 50th percentile (both the mean and median of the distribution), +1σthe 84.13th percentile, +2σthe 97.72nd percentile, and +3σthe 99.87th percentile. This is related to the68–95–99.7 ruleor the three-sigma rule. Note that in theory the 0th percentile falls at negative infinity and the 100th percentile at positive infinity, although in many practical applications, such as test results, natural lower and/or upper limits are enforced.

Applications

WhenISPsbill"burstable" internet bandwidth,the 95th or 98th percentile usually cuts off the top 5% or 2% of bandwidth peaks in each month, and then bills at the nearest rate. In this way, infrequent peaks are ignored, and the customer is charged in a fairer way. The reason this statistic is so useful in measuring data throughput is that it gives a very accurate picture of the cost of the bandwidth. The 95th percentile says that 95% of the time, the usage is below this amount: so, the remaining 5% of the time, the usage is above that amount.

Physicians will often use infant and children's weight and height to assess their growth in comparison to national averages and percentiles which are found ingrowth charts.

The 85th percentile speed of traffic on a road is often used as a guideline in settingspeed limitsand assessing whether such a limit is too high or low.

In finance,value at riskis a standard measure to assess (in a model-dependent way) the quantity under which the value of the portfolio is not expected to sink within a given period of time and given a confidence value.

Calculation methods

Interpolated and nearest-rank, exclusive and inclusive, percentiles for 10-score distribution

Interpolated and nearest-rank, exclusive and inclusive, percentiles for 10-score distribution

There are many formulas or algorithmsfor a percentile score. Hyndman and Fanidentified nine and most statistical and spreadsheet software use one of the methods they describe.Algorithms either return the value of a score that exists in the set of scores (nearest-rank methods) or interpolate between existing scores and are either exclusive or inclusive.

Nearest-rank methods (exclusive/inclusive)

PC: percentile specified	0.10	0.25	0.50	0.75	0.90
N: Number of scores	10	10	10	10	10
OR: ordinal rank = PC × N	1	2.5	5	7.5	9
Rank: >OR / ≥OR	2/1	3/3	6/5	8/8	10/9
Score at rank (exc/inc)	2/1	3/3	4/3	5/5	7/5

The figure shows a 10-score distribution, illustrates the percentile scores that result from these different algorithms, and serves as an introduction to the examples given subsequently. The simplest are nearest-rank methods that return a score from the distribution, although compared to interpolation methods, results can be a bit crude. The Nearest-Rank Methods table shows the computational steps for exclusive and inclusive methods.

Interpolated methods (exclusive/inclusive)

PC: percentile specified	0.10	0.25	0.50	0.75	0.90
N: number of scores	10	10	10	10	10
OR: PC×(N+1) / PC×(N−1)+1	1.1/1.9	2.75/3.25	5.5/5.5	8.25/7.75	9.9/9.1
LoRank: OR truncated	1/1	2/3	5/5	8/7	9/9
HIRank: OR rounded up	2/2	3/4	6/6	9/8	10/10
LoScore: score at LoRank	1/1	2/3	3/3	5/4	5/5
HiScore: score at HiRank	2/2	3/3	4/4	5/5	7/7
Difference: HiScore − LoScore	1/1	1/0	1/1	0/1	2/2
Mod: fractional part of OR	0.1/0.9	0.75/0.25	0.5/0.5	0.25/0.75	0.9/0.1
Interpolated score (exc/inc) = LoScore + Mod × Difference	1.1/1.9	2.75/3	3.5/3.5	5/4.75	6.8/5.2

Interpolation methods, as the name implies, can return a score that is between scores in the distribution. Algorithms used by statistical programs typically use interpolation methods, for example, the percentile.exc and percentile.inc functions in Microsoft Excel. The Interpolated Methods table shows the computational steps.

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 theP-th percentile$(0<P\leq 100)$of a list ofNordered values (sorted from least to greatest) is the smallest value in the list such that no more thanPpercent of the data is strictly less than the value and at leastPpercent 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. Theordinalranknis calculated using this formula

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

• 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 uselinear interpolationbetween adjacent ranks.

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

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

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

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

where$\lfloor x\rfloor$uses thefloor functionto represent the integral part of positivex,whereas$x{\bmod {1}}$uses themod functionto represent its fractional part (the remainder after division by 1). (Note that, though at the endpoint$x=N$,$v_{\lfloor x\rfloor +1}$is undefined, it does not need to be because it is multiplied by$x{\bmod {1}}=0$.) As we can see,xis the continuous version of the subscripti,linearly interpolatingvbetween adjacent nodes.

There are two ways in which the variant approaches differ. The first is in the linear relationship between therankx,thepercent rank$P=100p$,and a constant that is a function of the sample sizeN:

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

There is the additional requirement that the midpoint of the range$(1,N)$,corresponding to themedian,occur at$p=0.5$:

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

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

$x=f(p,N)=(N+1-2C)p+C.$

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

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

where

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

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

Furthermore, let

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

The inverse relationship is restricted to a narrower region:

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

Second variant,C= 1

[Source: Some software packages, includingNumPyandMicrosoft Excel(up to and including version 2013 by means of the PERCENTILE.INC function). Noted as an alternative byNIST.]

$x=f(p,N)=p(N-1)+1{\text{, }}p\in [0,1]$

$\therefore p={\frac {x-1}{N-1}}{\text{, }}x\in [1,N].$

Note that the$x\leftrightarrow p$relationship is one-to-one for$p\in [0,1]$,the only one of the three variants with this property; hence the "INC" suffix, forinclusive,on the Excel function.

Third variant,C= 0

(The primary variant recommended byNIST.Adopted by Microsoft Excel since 2010 by means of PERCENTIL.EXC function. However, as the "EXC" suffix indicates, the Excel versionexcludesboth endpoints of the range ofp,i.e.,$p\in (0,1)$,whereas the "INC" version, the second variant, does not; in fact, any number smaller than${\frac {1}{N+1}}$is also excluded and would cause an error.)

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

The inverse is restricted to a narrower region:

$p={\frac {x}{N+1}}{\text{, }}x\in (0,N).$

The weighted percentile method

See also:Weighted median

In addition to the percentile function, there is also aweighted 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$w_{1},w_{2},w_{3},\dots,w_{N}$associated, respectively, with ourNsorted sample values. Let

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

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

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

or

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

and

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

The 50% weighted percentile is known as theweighted median.

See also

• Decile
• Percentile rank
• Quantile
• Summary statistics