{"history":[],"article":{"id":"36d49089-8df5-489f-8f53-d18edb8d642a","title":"Quantile","slug":"quantile","content":":::summary\n:::\n\n# \nIqr with quantile.png, with quantiles shown. The area below the red curve is the same in the intervals\n(−∞,Q1)\n,\n(Q1,Q2)\n,\n(Q2,Q3)\n, and\n(Q3,+∞)\n.]]\nIn [statistics](https://bliptext.com/articles/statistics) and [probability](https://bliptext.com/articles/probability), **quantiles** are cut points dividing the [range](https://bliptext.com/articles/range-statistics) of a [probability](https://bliptext.com/articles/probability) distribution into continuous intervals with equal probabilities, or dividing the [observations](https://bliptext.com/articles/observation-statistics) in a [sample](https://bliptext.com/articles/sample-statistics) in the same way.\nThere is one fewer quantile than the number of groups created.\nCommon quantiles have special names, such as [quartiles](https://bliptext.com/articles/quartile) (four groups), [deciles](https://bliptext.com/articles/decile) (ten groups), and [percentiles](https://bliptext.com/articles/percentile) (100 groups).\nThe groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.\n**q**-**q**uantiles are values that [partition](https://bliptext.com/articles/partition-of-a-set) a [finite set](https://bliptext.com/articles/finite-set) of values into **q** [subsets](https://bliptext.com/articles/subset) of (nearly) e**q**ual sizes.\nThere are\nq − 1\npartitions of the q-quantiles, one for each [integer](https://bliptext.com/articles/integer) k satisfying\n0 < k < q\n. In some cases the value of a quantile may not be uniquely determined, as can be the case for the [median](https://bliptext.com/articles/median) (2-quantile) of a uniform probability distribution on a set of even size.\nQuantiles can also be applied to [continuous](https://bliptext.com/articles/continuous-function) distributions, providing a way to generalize [rank statistics](https://bliptext.com/articles/rank-statistics) to [continuous](https://bliptext.com/articles/continuous-function) variables (see [percentile rank](https://bliptext.com/articles/percentile-rank)).\nWhen the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function) of a [random variable](https://bliptext.com/articles/random-variable) is known, the q-quantiles are the application of the [quantile function](https://bliptext.com/articles/quantile-function) (the [inverse function](https://bliptext.com/articles/inverse-function) of the [cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function)) to the values\n{1/q, 2/q, …, (q − 1)/q}\n.\n\n# Quantiles of a population\nAs in the computation of, for example, [standard deviation](https://bliptext.com/articles/standard-deviation), the estimation of a quantile depends upon whether one is operating with a [statistical population](https://bliptext.com/articles/statistical-population) or with a [sample](https://bliptext.com/articles/sample-statistics) drawn from it.\nFor a population, of discrete values or for a continuous population density, the k-th q-quantile is the data value where the cumulative distribution function crosses\nk/q\n. That is, x is a k-th q-quantile for a variable X if\nPr[X < x] ≤ k/q\nor, equivalently,\nPr[X ≥ x] ≥ 1 − k/q\nand\nPr[X ≤ x] ≥ k/q\n.\nFor a finite population of N equally probable values indexed\n1, …, N\nfrom lowest to highest, the k-th q-quantile of this population can equivalently be computed via the value of.\nIf Ip is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the k-th q-quantile.\nOn the other hand, if Ip is an integer then any number from the data value at that index to the data value of the next index can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see [Estimating quantiles from a sample](https://bliptext.com/articles/)).\nIf, instead of using integers k and q, the \"p-quantile\" is based on a [real number](https://bliptext.com/articles/real-number) p with\n0 < p < 1\nthen p replaces\nk/q\nin the above formulas.\nThis broader terminology is used when quantiles are used to [parameterize continuous probability distributions](https://bliptext.com/articles/quantile-parameterized-distribution).\nMoreover, some software programs (including [Microsoft Excel](https://bliptext.com/articles/microsoft-excel)) regard the minimum and maximum as the 0th and 100th percentile, respectively.\nHowever, this broader terminology is an extension beyond traditional statistics definitions.\n\n## Examples\nThe following two examples use the Nearest Rank definition of quantile with rounding.\nFor an explanation of this definition, see [percentiles](https://bliptext.com/articles/percentile).\n\n### Even-sized population\nConsider an ordered population of 10 data values [3, 6, 7, 8, 8, 10, 13, 15, 16, 20].\nWhat are the 4-quantiles (the \"quartiles\") of this dataset?\nSo the first, second and third 4-quantiles (the \"quartiles\") of the dataset [3, 6, 7, 8, 8, 10, 13, 15, 16, 20] are [7, 9, 15].\nIf also required, the zeroth quartile is 3 and the fourth quartile is 20.\n\n### Odd-sized population\nConsider an ordered population of 11 data values [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20].\nWhat are the 4-quantiles (the \"quartiles\") of this dataset?\nSo the first, second and third 4-quantiles (the \"quartiles\") of the dataset [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20] are [7, 9, 15].\nIf also required, the zeroth quartile is 3 and the fourth quartile is 20.\n\n## Relationship to the mean\nFor any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that\nwhere Q(p) is the value of the p-quantile for\n0 < p < 1\n(or equivalently is the k-th q-quantile for\np = k/q\n), where μ is the distribution's [arithmetic mean](https://bliptext.com/articles/arithmetic-mean), and where σ is the distribution's [standard deviation](https://bliptext.com/articles/standard-deviation). In particular, the median\n(p = k/q = 1/2)\nis never more than one standard deviation from the mean.\nThe above formula can be used to bound the value\nμ + zσ\nin terms of quantiles.\nWhen\nz ≥ 0\n, the value that is [[standard score|\nz\nstandard deviations above the mean]] has a lower bound\nFor example, the value that is\nz = 1\nstandard deviation above the mean is always greater than or equal to\nQ(p = 0.5)\n, the median, and the value that is\nz = 2\nstandard deviations above the mean is always greater than or equal to\nQ(p = 0.8)\n, the fourth quintile.\nWhen\nz ≤ 0\n, there is instead an upper bound\nFor example, the value\nμ + zσ\nfor\nz = −3\nwill never exceed\nQ(p = 0.1)\n, the first decile.\n\n# Estimating quantiles from a sample\nOne problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size N.\nModern statistical packages rely on a number of techniques to [estimate](https://bliptext.com/articles/estimation-theory) the quantiles.\n[Hyndman](https://bliptext.com/articles/rob-j-hyndman) and Fan compiled a [taxonomy](https://bliptext.com/articles/taxonomy-general) of nine algorithms used by various software packages.\nAll methods compute Qp, the estimate for the p-quantile (the k-th q-quantile, where\n) from a sample of size N by computing a real valued index h. When h is an integer, the h-th smallest of the N values, xh, is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from h,\nx⌊h⌋\n, and\nx⌈h⌉\n. (For notation, see [floor and ceiling functions](https://bliptext.com/articles/floor-and-ceiling-functions)).\nThe first three are piecewise constant, changing abruptly at each data point, while the last six use linear interpolation between data points, and differ only in how the index h used to choose the point along the piecewise linear interpolation curve, is chosen.\n[Mathematica](https://bliptext.com/articles/mathematica), [Matlab](https://bliptext.com/articles/matlab), [R](https://bliptext.com/articles/r-programming-language) and [GNU Octave](https://bliptext.com/articles/gnu-octave) programming languages support all nine sample quantile methods.\n[SAS](https://bliptext.com/articles/sas-software) includes five sample quantile methods, [SciPy](https://bliptext.com/articles/scipy) and [Maple](https://bliptext.com/articles/maple-software) both include eight, [EViews](https://bliptext.com/articles/eviews) and [Julia](https://bliptext.com/articles/julia-programming-language) include the six piecewise linear functions, [Stata](https://bliptext.com/articles/stata) includes two, [Python](https://bliptext.com/articles/python-programming-language) includes two, and [Microsoft Excel](https://bliptext.com/articles/microsoft-excel) includes two.\nMathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods.\nThe estimate types and interpolation schemes used include:\nNotes:\nwhen\n.\np < 1/N\nand\np > (N − 1)/N\n. Choices include returning an error value, computing linear extrapolation, or assuming a constant value.\nOf the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as the default.\nThe [standard error](https://bliptext.com/articles/standard-error-statistics) of a quantile estimate can in general be estimated via the [bootstrap](https://bliptext.com/articles/bootstrap-statistics).\nThe Maritz–Jarrett method can also be used.\n\n# The asymptotic distribution of the sample median\nThe sample median is the most examined one amongst quantiles, being an alternative to estimate a location parameter, when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic.\nMoreover, the sample median is a more robust estimator than the sample mean.\nOne peculiarity of the sample median is its asymptotic distribution: when the sample comes from a continuous distribution, then the sample median has the anticipated Normal asymptotic distribution,\nThis extends to the other quantiles,\nwhere\nf(xp)\nis the value of the distribution density at the p-th population quantile.\nBut when the distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in https://stats.stackexchange.com/a/86638/28746).\nA solution to this problem is to use an alternative definition of sample quantiles through the concept of the \"mid-distribution\" function, which is defined as\nThe definition of sample quantiles through the concept of mid-distribution function can be seen as a generalization that can cover as special cases the continuous distributions.\nFor discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see Ma, Y., Genton, M. G., & Parzen, E. (2011).\nAsymptotic properties of sample quantiles of discrete distributions.\nAnnals of the Institute of Statistical Mathematics, 63(2), 227–243.\n\n# Approximate quantiles from a stream\nComputing approximate quantiles from data arriving from a stream can be done efficiently using compressed data structures.\nThe most popular methods are t-digest and KLL.\nThese methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile.\nBoth algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight.\nIf the stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles.\nWith more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles.\nSome values may be discarded from the stream and contribute to the weight of a nearby value without changing the quantile results too much.\nThe t-digest maintains a data structure of bounded size using an approach motivated by k-means clustering to group similar values.\nThe KLL algorithm uses a more sophisticated \"compactor\" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to p.\nBoth methods belong to the family of data sketches that are subsets of [Streaming Algorithms](https://bliptext.com/articles/streaming-algorithm) with useful properties: t-digest or KLL sketches can be combined.\nComputing the sketch for a very large vector of values can be split into trivially parallel processes where sketches are computed for partitions of the vector in parallel and merged later.\nThe algorithms described so far directly approximate the empirical quantiles without any particular assumptions on the data, in essence the data are simply numbers or more generally, a set of items that can be ordered.\nThese algorithms are computer science derived methods.\nAnother class of algorithms exist which assume that the data are realizations of a random process.\nThese are statistics derived methods, sequential nonparametric estimation algorithms in particular.\nThere are a number of such algorithms such as those based on stochastic approximation or Hermite series estimators.\nThese statistics based algorithms typically have constant update time and space complexity, but have different error bound guarantees compared to computer science type methods and make more assumptions.\nThe statistics based algorithms do present certain advantages however, particularly in the non-stationary streaming setting i.e. time-varying data.\nThe algorithms of both classes, along with some respective advantages and disadvantages have been recently surveyed.\n\n# Discussion\nStandardized test results are commonly reported as a student scoring \"in the 80th percentile\", for example.\nThis uses an alternative meaning of the word percentile as the interval between (in this case) the 80th and the 81st scalar percentile.\nThis separate meaning of percentile is also used in peer-reviewed scientific research articles.\nThe meaning used can be derived from its context.\nIf a distribution is symmetric, then the median is the mean (so long as the latter exists).\nBut, in general, the median and the mean can differ.\nFor instance, with a random variable that has an [exponential distribution](https://bliptext.com/articles/exponential-distribution), any particular sample of this random variable will have roughly a 63% chance of being less than the mean.\nThis is because the exponential distribution has a long tail for positive values but is zero for negative numbers.\nQuantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers.\nEmpirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from the mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics.\nClosely related is the subject of [least absolute deviations](https://bliptext.com/articles/least-absolute-deviations), a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error.\nThe connection is that the mean is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error.\n[Least absolute deviations](https://bliptext.com/articles/least-absolute-deviations) shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of [robust regression](https://bliptext.com/articles/robust-regression) are available.\nThe quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if m is the median of a random variable X, then\n2m\nis the median of\n2X\n, unless an arbitrary choice has been made from a range of values to specify a particular quantile.\n(See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only [ordinal](https://bliptext.com/articles/ordinal-scale) data are available.\n\n# Other quantifications\nValues that divide sorted data into equal subsets other than four have different names.\nQ3 − Q1\n.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:36:30.628Z","updated_at":"2025-01-25T16:36:30.628Z","search_vector":"'quantil':1","search_content":" \n \n\n# \nIqr with quantile.png, with quantiles shown. The area below the red curve is the same in the intervals\n(−∞,Q1)\n,\n(Q1,Q2)\n,\n(Q2,Q3)\n, and\n(Q3,+∞)\n.]]\nIn [statistics](https://bliptext.com/articles/statistics) and [probability](https://bliptext.com/articles/probability), **quantiles** are cut points dividing the [range](https://bliptext.com/articles/range-statistics) of a [probability](https://bliptext.com/articles/probability) distribution into continuous intervals with equal probabilities, or dividing the [observations](https://bliptext.com/articles/observation-statistics) in a [sample](https://bliptext.com/articles/sample-statistics) in the same way.\nThere is one fewer quantile than the number of groups created.\nCommon quantiles have special names, such as [quartiles](https://bliptext.com/articles/quartile) (four groups), [deciles](https://bliptext.com/articles/decile) (ten groups), and [percentiles](https://bliptext.com/articles/percentile) (100 groups).\nThe groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.\n**q**-**q**uantiles are values that [partition](https://bliptext.com/articles/partition-of-a-set) a [finite set](https://bliptext.com/articles/finite-set) of values into **q** [subsets](https://bliptext.com/articles/subset) of (nearly) e**q**ual sizes.\nThere are\nq − 1\npartitions of the q-quantiles, one for each [integer](https://bliptext.com/articles/integer) k satisfying\n0 < k < q\n. In some cases the value of a quantile may not be uniquely determined, as can be the case for the [median](https://bliptext.com/articles/median) (2-quantile) of a uniform probability distribution on a set of even size.\nQuantiles can also be applied to [continuous](https://bliptext.com/articles/continuous-function) distributions, providing a way to generalize [rank statistics](https://bliptext.com/articles/rank-statistics) to [continuous](https://bliptext.com/articles/continuous-function) variables (see [percentile rank](https://bliptext.com/articles/percentile-rank)).\nWhen the [[cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function)](https://bliptext.com/articles/cumulative-distribution-function) of a [random variable](https://bliptext.com/articles/random-variable) is known, the q-quantiles are the application of the [quantile function](https://bliptext.com/articles/quantile-function) (the [inverse function](https://bliptext.com/articles/inverse-function) of the [[cumulative distribution function](https://bliptext.com/articles/cumulative-distribution-function)](https://bliptext.com/articles/cumulative-distribution-function)) to the values\n{1/q, 2/q, …, (q − 1)/q}\n.\n\n# Quantiles of a population\nAs in the computation of, for example, [standard deviation](https://bliptext.com/articles/standard-deviation), the estimation of a quantile depends upon whether one is operating with a [statistical population](https://bliptext.com/articles/statistical-population) or with a [sample](https://bliptext.com/articles/sample-statistics) drawn from it.\nFor a population, of discrete values or for a continuous population density, the k-th q-quantile is the data value where the cumulative distribution function crosses\nk/q\n. That is, x is a k-th q-quantile for a variable X if\nPr[X < x] ≤ k/q\nor, equivalently,\nPr[X ≥ x] ≥ 1 − k/q\nand\nPr[X ≤ x] ≥ k/q\n.\nFor a finite population of N equally probable values indexed\n1, …, N\nfrom lowest to highest, the k-th q-quantile of this population can equivalently be computed via the value of.\nIf Ip is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the k-th q-quantile.\nOn the other hand, if Ip is an integer then any number from the data value at that index to the data value of the next index can be taken as the quantile, and it is conventional (though arbitrary) to take the average of those two values (see [Estimating quantiles from a sample](https://bliptext.com/articles/)).\nIf, instead of using integers k and q, the \"p-quantile\" is based on a [real number](https://bliptext.com/articles/real-number) p with\n0 < p < 1\nthen p replaces\nk/q\nin the above formulas.\nThis broader terminology is used when quantiles are used to [parameterize continuous probability distributions](https://bliptext.com/articles/quantile-parameterized-distribution).\nMoreover, some software programs (including [Microsoft Excel](https://bliptext.com/articles/microsoft-excel)) regard the minimum and maximum as the 0th and 100th percentile, respectively.\nHowever, this broader terminology is an extension beyond traditional statistics definitions.\n\n## Examples\nThe following two examples use the Nearest Rank definition of quantile with rounding.\nFor an explanation of this definition, see [percentiles](https://bliptext.com/articles/percentile).\n\n### Even-sized population\nConsider an ordered population of 10 data values [3, 6, 7, 8, 8, 10, 13, 15, 16, 20].\nWhat are the 4-quantiles (the \"quartiles\") of this dataset?\nSo the first, second and third 4-quantiles (the \"quartiles\") of the dataset [3, 6, 7, 8, 8, 10, 13, 15, 16, 20] are [7, 9, 15].\nIf also required, the zeroth quartile is 3 and the fourth quartile is 20.\n\n### Odd-sized population\nConsider an ordered population of 11 data values [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20].\nWhat are the 4-quantiles (the \"quartiles\") of this dataset?\nSo the first, second and third 4-quantiles (the \"quartiles\") of the dataset [3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20] are [7, 9, 15].\nIf also required, the zeroth quartile is 3 and the fourth quartile is 20.\n\n## Relationship to the mean\nFor any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that\nwhere Q(p) is the value of the p-quantile for\n0 < p < 1\n(or equivalently is the k-th q-quantile for\np = k/q\n), where μ is the distribution's [arithmetic mean](https://bliptext.com/articles/arithmetic-mean), and where σ is the distribution's [standard deviation](https://bliptext.com/articles/standard-deviation). In particular, the median\n(p = k/q = 1/2)\nis never more than one standard deviation from the mean.\nThe above formula can be used to bound the value\nμ + zσ\nin terms of quantiles.\nWhen\nz ≥ 0\n, the value that is [[standard score|\nz\nstandard deviations above the mean]] has a lower bound\nFor example, the value that is\nz = 1\nstandard deviation above the mean is always greater than or equal to\nQ(p = 0.5)\n, the median, and the value that is\nz = 2\nstandard deviations above the mean is always greater than or equal to\nQ(p = 0.8)\n, the fourth quintile.\nWhen\nz ≤ 0\n, there is instead an upper bound\nFor example, the value\nμ + zσ\nfor\nz = −3\nwill never exceed\nQ(p = 0.1)\n, the first decile.\n\n# Estimating quantiles from a sample\nOne problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size N.\nModern statistical packages rely on a number of techniques to [estimate](https://bliptext.com/articles/estimation-theory) the quantiles.\n[Hyndman](https://bliptext.com/articles/rob-j-hyndman) and Fan compiled a [taxonomy](https://bliptext.com/articles/taxonomy-general) of nine algorithms used by various software packages.\nAll methods compute Qp, the estimate for the p-quantile (the k-th q-quantile, where\n) from a sample of size N by computing a real valued index h. When h is an integer, the h-th smallest of the N values, xh, is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from h,\nx⌊h⌋\n, and\nx⌈h⌉\n. (For notation, see [floor and ceiling functions](https://bliptext.com/articles/floor-and-ceiling-functions)).\nThe first three are piecewise constant, changing abruptly at each data point, while the last six use linear interpolation between data points, and differ only in how the index h used to choose the point along the piecewise linear interpolation curve, is chosen.\n[Mathematica](https://bliptext.com/articles/mathematica), [Matlab](https://bliptext.com/articles/matlab), [R](https://bliptext.com/articles/r-programming-language) and [GNU Octave](https://bliptext.com/articles/gnu-octave) programming languages support all nine sample quantile methods.\n[SAS](https://bliptext.com/articles/sas-software) includes five sample quantile methods, [SciPy](https://bliptext.com/articles/scipy) and [Maple](https://bliptext.com/articles/maple-software) both include eight, [EViews](https://bliptext.com/articles/eviews) and [Julia](https://bliptext.com/articles/julia-programming-language) include the six piecewise linear functions, [Stata](https://bliptext.com/articles/stata) includes two, [Python](https://bliptext.com/articles/python-programming-language) includes two, and [Microsoft Excel](https://bliptext.com/articles/microsoft-excel) includes two.\nMathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods.\nThe estimate types and interpolation schemes used include:\nNotes:\nwhen\n.\np < 1/N\nand\np > (N − 1)/N\n. Choices include returning an error value, computing linear extrapolation, or assuming a constant value.\nOf the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as the default.\nThe [standard error](https://bliptext.com/articles/standard-error-statistics) of a quantile estimate can in general be estimated via the [bootstrap](https://bliptext.com/articles/bootstrap-statistics).\nThe Maritz–Jarrett method can also be used.\n\n# The asymptotic distribution of the sample median\nThe sample median is the most examined one amongst quantiles, being an alternative to estimate a location parameter, when the expected value of the distribution does not exist, and hence the sample mean is not a meaningful estimator of a population characteristic.\nMoreover, the sample median is a more robust estimator than the sample mean.\nOne peculiarity of the sample median is its asymptotic distribution: when the sample comes from a continuous distribution, then the sample median has the anticipated Normal asymptotic distribution,\nThis extends to the other quantiles,\nwhere\nf(xp)\nis the value of the distribution density at the p-th population quantile.\nBut when the distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in https://stats.stackexchange.com/a/86638/28746).\nA solution to this problem is to use an alternative definition of sample quantiles through the concept of the \"mid-distribution\" function, which is defined as\nThe definition of sample quantiles through the concept of mid-distribution function can be seen as a generalization that can cover as special cases the continuous distributions.\nFor discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see Ma, Y., Genton, M. G., & Parzen, E. (2011).\nAsymptotic properties of sample quantiles of discrete distributions.\nAnnals of the Institute of Statistical Mathematics, 63(2), 227–243.\n\n# Approximate quantiles from a stream\nComputing approximate quantiles from data arriving from a stream can be done efficiently using compressed data structures.\nThe most popular methods are t-digest and KLL.\nThese methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a specified quantile.\nBoth algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight.\nIf the stream is made of a repetition of 100 times v1 and 100 times v2, there is no reason to keep a sorted list of 200 elements, it is enough to keep two elements and two counts to be able to recover the quantiles.\nWith more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles.\nSome values may be discarded from the stream and contribute to the weight of a nearby value without changing the quantile results too much.\nThe t-digest maintains a data structure of bounded size using an approach motivated by k-means clustering to group similar values.\nThe KLL algorithm uses a more sophisticated \"compactor\" method that leads to better control of the error bounds at the cost of requiring an unbounded size if errors must be bounded relative to p.\nBoth methods belong to the family of data sketches that are subsets of [Streaming Algorithms](https://bliptext.com/articles/streaming-algorithm) with useful properties: t-digest or KLL sketches can be combined.\nComputing the sketch for a very large vector of values can be split into trivially parallel processes where sketches are computed for partitions of the vector in parallel and merged later.\nThe algorithms described so far directly approximate the empirical quantiles without any particular assumptions on the data, in essence the data are simply numbers or more generally, a set of items that can be ordered.\nThese algorithms are computer science derived methods.\nAnother class of algorithms exist which assume that the data are realizations of a random process.\nThese are statistics derived methods, sequential nonparametric estimation algorithms in particular.\nThere are a number of such algorithms such as those based on stochastic approximation or Hermite series estimators.\nThese statistics based algorithms typically have constant update time and space complexity, but have different error bound guarantees compared to computer science type methods and make more assumptions.\nThe statistics based algorithms do present certain advantages however, particularly in the non-stationary streaming setting i.e. time-varying data.\nThe algorithms of both classes, along with some respective advantages and disadvantages have been recently surveyed.\n\n# Discussion\nStandardized test results are commonly reported as a student scoring \"in the 80th percentile\", for example.\nThis uses an alternative meaning of the word percentile as the interval between (in this case) the 80th and the 81st scalar percentile.\nThis separate meaning of percentile is also used in peer-reviewed scientific research articles.\nThe meaning used can be derived from its context.\nIf a distribution is symmetric, then the median is the mean (so long as the latter exists).\nBut, in general, the median and the mean can differ.\nFor instance, with a random variable that has an [exponential distribution](https://bliptext.com/articles/exponential-distribution), any particular sample of this random variable will have roughly a 63% chance of being less than the mean.\nThis is because the exponential distribution has a long tail for positive values but is zero for negative numbers.\nQuantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers.\nEmpirically, if the data being analyzed are not actually distributed according to an assumed distribution, or if there are other potential sources for outliers that are far removed from the mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics.\nClosely related is the subject of [least absolute deviations](https://bliptext.com/articles/least-absolute-deviations), a method of regression that is more robust to outliers than is least squares, in which the sum of the absolute value of the observed errors is used in place of the squared error.\nThe connection is that the mean is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error.\n[Least absolute deviations](https://bliptext.com/articles/least-absolute-deviations) shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of [robust regression](https://bliptext.com/articles/robust-regression) are available.\nThe quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if m is the median of a random variable X, then\n2m\nis the median of\n2X\n, unless an arbitrary choice has been made from a range of values to specify a particular quantile.\n(See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only [ordinal](https://bliptext.com/articles/ordinal-scale) data are available.\n\n# Other quantifications\nValues that divide sorted data into equal subsets other than four have different names.\nQ3 − Q1\n.","revisionCount":0},"hasMore":false,"nextCursor":null}

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