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ions 4 calculation methods toggle calculation methods subsection 4 1 the nearest rank method 4 2 the cdf method 4 3 the linear interpolation between closest ranks method 4 3 1 first variant c 1 2 4 3 2 second variant c 1 4 3 3 third variant c 0 4 4 the weighted percentile method 5 see also 6 notes 7 references toggle the table of contents percentile 27 languages العربية čeština dansk deutsch español euskara فارسی français galego עברית italiano 日本語 한국어 nederlands norsk bokmål polski português русский simple english српски srpski svenska ไทย türkçe українська اردو 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia statistic which divides a data set into 100 parts and analyzes it as a percentage in statistics a percentile or percentile score also known as centile often denoted as p k displaystyle p_ k or pk for a given percentage k 1 2 is a score greater than a given percentage of all scores or data point value in a sample a i e a score in the k th percentile would be above approximately k of all scores in its sample for example the 97th percentile p97 is the value such that 97 of the data points are less than it percentiles are a type of quantiles obtained by a subdivision into 100 groups the 25th percentile p25 is also known as the first quartile q 1 the 50th percentile p50 as the median or second quartile q 2 and the 75th percentile p75 as the third quartile q 3 for example the 50th percentile median is the score below or at or below depending on the definition which 50 of the scores in the distribution are found percentiles are expressed in the same unit of measurement as the input scores not in percent for example if the scores refer to human weight the corresponding percentiles will be expressed in kilograms or pounds in the limit of an infinite sample size the percentile approximates the percentile function the inverse of the cumulative distribution function a related quantity is the percentile rank of a given score expressed as a percentage 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 of test scores from norm 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 edit there is no standard definition of percentile 3 4 5 however all definitions yield similar results when the number of observations is very large and the probability distribution is continuous 6 in the limit as the sample size approaches infinity the 100 p th percentile 0 p 1 approximates the inverse of the cumulative distribution function cdf thus formed evaluated at p as p approximates the cdf this can be seen as a consequence of the glivenko cantelli theorem some methods for calculating the percentiles are given below in the normal distribution edit representation of the three sigma rule the dark blue zone represents observations within one standard deviation σ to either side of the mean 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 the calculation methods section below are approximations for use in small sample statistics in general terms for very large populations following a normal distribution percentiles may often be represented by reference to a normal curve plot the normal distribution is plotted along an axis scaled to standard deviations or sigma σ displaystyle sigma units mathematically the normal distribution extends to negative infinity 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 3 σ to 3 σ range for example with human heights very few people are above the 3 σ 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 3 σ is the 0 13th percentile 2 σ the 2 28th percentile 1 σ the 15 87th percentile 0 σ the 50th percentile both the mean and median of the distribution 1 σ the 84 13th percentile 2 σ the 97 72nd percentile and 3 σ the 99 87th percentile this is related to the 68 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 edit when internet service providers 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 median and other percentiles which are found in growth charts the 85th percentile speed of traffic on a road is often used as a guideline in setting speed limits and assessing whether such a limit is too high or low 7 8 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 edit this section possibly contains original synthesis source material should verifiably mention and relate to the main topic relevant discussion may be found on the talk page february 2023 learn how and when to remove this message 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 9 for a percentile score hyndman and fan 3 identified nine and most statistical and spreadsheet software use one of the methods they describe 10 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 edit 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 0 p 100 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 at least p percent of the data is less than or equal to that value this is calculated 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 the formula n p 100 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 this method is also called the empirical distribution function method 11 the 50th percentile calculated using this method is equal to the usual value of the median if n is odd but not if n is even 11 the cdf method edit the cdf method was described e g by langford 11 given the order statistics v i i 1 2 n v i 1 v i 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 calculate x p 100 n displaystyle x frac p 100 times n the percentile is v x v 1 for x 0 v x for x 0 1 2 n 1 v x v x 1 2 for x 1 2 n 1 displaystyle v x begin cases v_ 1 text for x 0 v_ lceil x rceil text for x notin 0 1 2 ldots n 1 frac v_ x v_ x 1 2 text for x in 1 2 ldots n 1 end cases the 50th percentile calculated using this method is equal to the usual value of the median doubling the data leaves the calculated percentile value unchanged cdf percentile values for ordered list 15 20 35 40 50 n 5 displaystyle n 5 percentage p displaystyle p 0 p 20 displaystyle 0 leq p 20 p 20 displaystyle p 20 20 p 40 displaystyle 20 p 40 p 40 displaystyle p 40 40 p 60 displaystyle 40 p 60 p 60 displaystyle p 60 60 p 80 displaystyle 60 p 80 p 80 displaystyle p 80 80 p 100 displaystyle 80 p leq 100 x p 100 n displaystyle x frac p 100 times n 0 x 1 displaystyle 0 leq x 1 x 1 displaystyle x 1 1 x 2 displaystyle 1 x 2 x 2 displaystyle x 2 2 x 3 displaystyle 2 x 3 x 3 displaystyle x 3 3 x 4 displaystyle 3 x 4 x 4 displaystyle x 4 4 x 5 displaystyle 4 x leq 5 percentile 15 17 5 20 27 5 35 37 5 40 45 50 the linear interpolation between closest ranks method edit an alternative to rounding used in many applications is to use linear interpolation between adjacent ranks given the order statistics v i i 1 2 n v i 1 v i 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 the linear interpolation is simply accomplished by calculating v x v 1 for x 1 v n for x n v x x mod 1 v x 1 v x for 1 x n displaystyle v x begin cases v_ 1 text for x leq 1 v_ n text for x n v_ lfloor x rfloor x bmod 1 v_ lfloor x rfloor 1 v_ lfloor x rfloor text for 1 x n end cases where x displaystyle lfloor x rfloor uses the floor function to represent the integral part of x whereas x mod 1 displaystyle x bmod 1 uses the mod function to represent its fractional part the remainder after division by 1 this way v i v i for i 1 2 n displaystyle v i v_ i text for i 1 2 ldots n 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 p 100 p displaystyle p 100p and a constant that is a function of the sample size n x f p n n c 1 p c 2 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 the median occur at p 0 5 displaystyle p 0 5 f 0 5 n n c 1 2 c 2 n 1 2 2 c 2 c 1 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 2 c 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 of p 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 of c 1 2 1 ξ 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 edit the result of using each of the three variants on the ordered list 15 20 35 40 50 sources matlab prctile function 12 11 x f p n p 1 2 p p 1 p n 1 p 0 p 1 n p p n 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 p i 1 n i 1 2 i 1 n n displaystyle p_ i frac 1 n left i frac 1 2 right i in 1 n cap mathbb n p 1 1 2 n p n 2 n 1 2 n displaystyle therefore p_ 1 frac 1 2n p_ n frac 2n 1 2n furthermore let p i 100 p i displaystyle p_ i 100p_ i the inverse relationship is restricted to a narrower region p 1 n x 1 2 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 edit source some software packages including numpy 13 and microsoft excel 5 up to and including version 2013 by means of the percentile inc function noted as an alternative by nist 10 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 1 n 1 x 1 n displaystyle therefore p frac x 1 n 1 text x in 1 n note that the x p displaystyle x leftrightarrow p relationship is one to one for p 0 1 displaystyle p in 0 1 the only one of the three variants with this property hence the inc suffix for inclusive on the excel function third variant c 0 edit the primary variant recommended by nist 10 adopted by microsoft excel since 2010 by means of percentil exc function however as the exc suffix indicates the excel version excludes both endpo...