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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 endpoints of the range of p i e p 0 1 displaystyle p in 0 1 whereas the inc version the second variant does not in fact any number smaller than 1 n 1 displaystyle frac 1 n 1 is also excluded and would cause an error x f p n 1 p 0 1 n 1 p n 1 p 1 n 1 n n 1 n p n n 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 x n 1 x 0 n displaystyle p frac x n 1 text x in 0 n the weighted percentile method edit see also weighted median 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 w 1 w 2 w 3 w n displaystyle w_ 1 w_ 2 w_ 3 dots w_ n associated respectively with our n sorted sample values let s n k 1 n w k displaystyle s_ n sum _ k 1 n w_ k the sum of the weights then the formulas above are generalized by taking p n 1 s n s n w n 2 displaystyle p_ n frac 1 s_ n left s_ n frac w_ n 2 right when c 1 2 displaystyle c 1 2 or p n s n c w n s n 1 2 c w n displaystyle p_ n frac s_ n cw_ n s_ n 1 2c w_ n for general c displaystyle c and v v k p p k p k 1 p k v k 1 v k 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 see also edit mathematics portal decile percentile rank quantile sample maximum and minimum summary statistics notes edit the typical definition based on greater than may be called exclusive in contrast to the alternative definition involving greater than or equal to so called because it includes one additional data point references edit stephens larry j 2011 04 12 schaum s outline of statistics for engineers mcgraw hill professional isbn 978 0 07 173647 3 couallier vincent gerville réache léo huber carol catherine limnios nikolaos mesbah mounir 2013 12 31 statistical models and methods for reliability and survival analysis john wiley sons isbn 978 1 84821 619 8 1 2 hyndman 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 lane david percentiles retrieved 2007 09 15 1 2 pottel hans statistical flaws in excel pdf archived from the original pdf on 2013 06 04 retrieved 2013 03 25 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 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 rational speed limits and the 85th percentile speed pdf lsp org louisiana state police archived from the original pdf on 23 september 2018 retrieved 28 october 2018 wessa p 2021 percentiles in free statistics software office for research development and education retrieved 13 november 2021 1 2 3 engineering statistics handbook percentile nist retrieved 2009 02 18 1 2 3 4 langford e 2006 quartiles in elementary statistics journal of statistics education 14 3 doi 10 1080 10691898 2006 11910589 matlab statistics toolbox percentiles retrieved 2006 09 15 this is equivalent to method 5 discussed here numpy 1 12 documentation scipy retrieved 2017 03 19 v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova anova analysis of covariance multivariate anova degrees of freedom categorical multivariate time series survival analysis categori...
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