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Percentile

From Wikipedia, the free encyclopedia
Statistic which divides a data set into 100 parts and analyzes it as a percentage

Instatistics, ak-thpercentile, also known aspercentile score orcentile, is ascore (e.g., a data point)below which a givenpercentagek of all arranged scores in itsfrequency distribution exists ("exclusive" definition) or a scoreat or below which a given percentage of the all arranged scores exists ("inclusive" definition); i.e. a score in thek-th percentile would be above approximatelyk% of all scores in its set. For example, the 97th percentile of data is a data point below which 97% of all data points exist (by the exclusive definition). Percentiles depends on how scores are arranged.

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 themedian or 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.

Percentiles are expressed in the sameunit of measurement as the input scores,not inpercent; for example, if the scores refer tohuman weight, the corresponding percentiles will be expressed in kilograms or pounds.In thelimit of an infinitesample size, the percentile approximates thepercentile function, the inverse of thecumulative distribution function.

A related quantity is thepercentile rank of 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 scores fromnorm-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

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There is no standard definition of percentile;[1][2][3]however, all definitions yield similar results when the number of observations is very large and theprobability distribution is continuous.[4] In the limit, as the sample size approaches infinity, the 100pth percentile (0<p<1) approximates the inverse of thecumulative distribution function (CDF) thus formed, evaluated atp, asp approximates 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

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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 methods section (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 (σ{\displaystyle \sigma }) units. Mathematically, the normal distribution extends to negativeinfinity on 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 −3σ to +3σ range. For example, with human heights very few people are above the +3σ height 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, −3σ is the 0.13th percentile, −2σ the 2.28th percentile, −1σ the 15.87th percentile, 0σ the 50th percentile (both the mean and median of the distribution), +1σ the 84.13th percentile, +2σ the 97.72nd percentile, and +3σ the 99.87th percentile. This is related to the68–95–99.7 rule or 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

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WhenISPs bill"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 limits and assessing whether such a limit is too high or low.[5][6]

In finance,value at risk is 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

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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 algorithms[7] for a percentile score. Hyndman and Fan[1] identified nine and most statistical and spreadsheet software use one of the methods they describe.[8] 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 specified0.100.250.500.750.90
N: Number of scores1010101010
OR: ordinal rank = PC × N12.557.59
Rank: >OR / ≥OR2/13/36/58/810/9
Score at rank (exc/inc)2/13/34/35/57/5

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 specified0.100.250.500.750.90
N: number of scores1010101010
OR: PC×(N+1) / PC×(N−1)+11.1/1.92.75/3.255.5/5.58.25/7.759.9/9.1
LoRank: OR truncated1/12/35/58/79/9
HIRank: OR rounded up2/23/46/69/810/10
LoScore: score at LoRank1/12/33/35/45/5
HiScore: score at HiRank2/23/34/45/57/7
Difference: HiScore − LoScore1/11/01/10/12/2
Mod: fractional part of OR0.1/0.90.75/0.250.5/0.50.25/0.750.9/0.1
Interpolated score (exc/inc)
= LoScore + Mod × Difference		1.1/1.92.75/33.5/3.55/4.756.8/5.2

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

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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≤100){\displaystyle (0<P\leq 100)} of a list ofN ordered values (sorted from least to greatest) is the smallest value in the list such that no more thanP percent of the data is strictly less than the value and at leastP 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. Theordinalrankn is calculated using this formula

n=⌈P100×N⌉.{\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

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

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

{vi,i=1,2,…,N:vi+1≥vi,∀i=1,2,…,N−1},{\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(vi,i){\displaystyle (v_{i},i)}. This is simply accomplished by

v(x)=v⌊x⌋+(xmod1)(v⌊x⌋+1−v⌊x⌋),∀x∈[1,N]:v(i)=vi, for i=1,2,…,N,{\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⌊x⌋{\displaystyle \lfloor x\rfloor } uses thefloor function to represent the integral part of positivex, whereasxmod1{\displaystyle x{\bmod {1}}} uses themod function to represent its fractional part (the remainder after division by 1). (Note that, though at the endpointx=N{\displaystyle x=N},v⌊x⌋+1{\displaystyle v_{\lfloor x\rfloor +1}} is undefined, it does not need to be because it is multiplied byxmod1=0{\displaystyle x{\bmod {1}}=0}.) As we can see,x is the continuous version of the subscripti, linearly interpolatingv between adjacent nodes.

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

x=f(p,N)=(N+c1)p+c2.{\displaystyle x=f(p,N)=(N+c_{1})p+c_{2}.}

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

f(0.5,N)=N+c12+c2=N+12∴2c2+c1=1,{\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:

x=f(p,N)=(N+1−2C)p+C.{\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[0,1]{\displaystyle [0,1]} range ofp:f(p,N){\displaystyle f(p,N)} should produce, or be forced to produce, a result in the range[1,N]{\displaystyle [1,N]}, which may mean the absence of a one-to-one correspondence in the wider region. One author has suggested a choice ofC=12(1+ξ){\displaystyle C={\tfrac {1}{2}}(1+\xi )} whereξ is the shape of theGeneralized extreme value distribution which is the extreme value limit of the sampled distribution.

First variant,C = 1/2

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

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

x=f(p)={Np+12,∀p∈[p1,pN],1,∀p∈[0,p1],N,∀p∈[pN,1].{\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

pi=1N(i−12),i∈[1,N]∩N{\displaystyle p_{i}={\frac {1}{N}}\left(i-{\frac {1}{2}}\right),i\in [1,N]\cap \mathbb {N} }
∴p1=12N,pN=2N−12N.{\displaystyle \therefore p_{1}={\frac {1}{2N}},p_{N}={\frac {2N-1}{2N}}.}

Furthermore, let

Pi=100pi.{\displaystyle P_{i}=100p_{i}.}

The inverse relationship is restricted to a narrower region:

p=1N(x−12),x∈(1,N)∩R.{\displaystyle p={\frac {1}{N}}\left(x-{\frac {1}{2}}\right),x\in (1,N)\cap \mathbb {R} .}

Second variant,C = 1

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[Source: Some software packages, includingNumPy[11] andMicrosoft Excel[3] (up to and including version 2013 by means of the PERCENTILE.INC function). Noted as an alternative byNIST.[8]]

x=f(p,N)=p(N−1)+1, p∈[0,1]{\displaystyle x=f(p,N)=p(N-1)+1{\text{, }}p\in [0,1]}
∴p=x−1N−1, x∈[1,N].{\displaystyle \therefore p={\frac {x-1}{N-1}}{\text{, }}x\in [1,N].}

Note that thex↔p{\displaystyle x\leftrightarrow p} relationship is one-to-one forp∈[0,1]{\displaystyle 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

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(The primary variant recommended byNIST.[8] Adopted by Microsoft Excel since 2010 by means of PERCENTIL.EXC function. However, as the "EXC" suffix indicates, the Excel versionexcludes both endpoints of the range ofp, i.e.,p∈(0,1){\displaystyle p\in (0,1)}, whereas the "INC" version, the second variant, does not; in fact, any number smaller than1N+1{\displaystyle {\frac {1}{N+1}}} is also excluded and would cause an error.)

x=f(p,N)={1, p∈[0,1N+1]p(N+1), p∈(1N+1,NN+1)N, p∈[NN+1,1].{\displaystyle 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=xN+1, x∈(0,N).{\displaystyle p={\frac {x}{N+1}}{\text{, }}x\in (0,N).}

The weighted percentile method

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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 weightsw1,w2,w3,…,wN{\displaystyle w_{1},w_{2},w_{3},\dots ,w_{N}} associated, respectively, with ourN sorted sample values. Let

SN=∑k=1Nwk,{\displaystyle S_{N}=\sum _{k=1}^{N}w_{k},}

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

pn=1SN(Sn−wn2){\displaystyle p_{n}={\frac {1}{S_{N}}}\left(S_{n}-{\frac {w_{n}}{2}}\right)} whenC=1/2{\displaystyle C=1/2},

or

pn=Sn−CwnSN+(1−2C)wn{\displaystyle p_{n}={\frac {S_{n}-Cw_{n}}{S_{N}+(1-2C)w_{n}}}} for generalC{\displaystyle C},

and

v=vk+P−pkpk+1−pk(vk+1−vk).{\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 theweighted median.

See also

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References

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  1. ^abHyndman, Rob J.; Fan, Yanan (November 1996)."Sample Quantiles in Statistical Packages".American Statistician.50 (4). American Statistical Association:361–365.doi:10.2307/2684934.JSTOR 2684934.
  2. ^Lane, David."Percentiles". Retrieved2007-09-15.
  3. ^abPottel, Hans."Statistical flaws in Excel"(PDF). Archived fromthe original(PDF) on 2013-06-04. Retrieved2013-03-25.
  4. ^Schoonjans F, De Bacquer D, Schmid P (2011)."Estimation of population percentiles".Epidemiology.22 (5):750–751.doi:10.1097/EDE.0b013e318225c1de.PMC 3171208.PMID 21811118.
  5. ^Johnson, Robert; Kuby, Patricia (2007), "Applied Example 2.15, The 85th Percentile Speed Limit: Going With 85% of the Flow",Elementary Statistics (10th ed.), Cengage Learning, p. 102,ISBN 9781111802493.
  6. ^"Rational Speed Limits and the 85th Percentile Speed"(PDF).lsp.org. Louisiana State Police. Archived fromthe original(PDF) on 23 September 2018. Retrieved28 October 2018.
  7. ^Wessa, P (2021)."Percentiles in Free Statistics Software". Office for Research Development and Education. Retrieved13 November 2021.
  8. ^abc"Engineering Statistics Handbook: Percentile".NIST. Retrieved2009-02-18.
  9. ^"Matlab Statistics Toolbox – Percentiles". Retrieved2006-09-15., This is equivalent to Method 5 discussedhere
  10. ^Langford, E. (2006)."Quartiles in Elementary Statistics".Journal of Statistics Education.14 (3).doi:10.1080/10691898.2006.11910589.
  11. ^"NumPy 1.12 documentation".SciPy. Retrieved2017-03-19.
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