20:
2203:
and two counts to be able to recover the quantiles. With more values, these algorithms maintain a trade-off between the number of unique values stored and the precision of the resulting quantiles. Some values may be discarded from the stream and contribute to the weight of a nearby value without changing the quantile results too much. The t-digest maintains a data structure of bounded size using an approach motivated by
3257:
2262:, 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. The connection is that the mean is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error.
2189:
The 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. For discrete distributions the sample median as defined through this concept has an asymptotically Normal distribution, see
615:
The second quartile value (same as the median) is determined by 11×(2/4) = 5.5, which rounds up to 6. Therefore, 6 is the rank in the population (from least to greatest values) at which approximately 2/4 of the values are less than the value of the second quartile (or median). The sixth value in
638:
Although not universally accepted, one can also speak of the fourth quartile. This is the maximum value of the set, so the fourth quartile in this example would be 20. Under the
Nearest Rank definition of quantile, the rank of the fourth quartile is the rank of the biggest number, so the rank of the
554:
Although not universally accepted, one can also speak of the fourth quartile. This is the maximum value of the set, so the fourth quartile in this example would be 20. Under the
Nearest Rank definition of quantile, the rank of the fourth quartile is the rank of the biggest number, so the rank of the
2202:
Both algorithms are based on a similar idea: compressing the stream of values by summarizing identical or similar values with a weight. If 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
2254:
Quantiles are useful measures because they are less susceptible than means to long-tailed distributions and outliers. Empirically, 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
2227:
algorithms exist which assume that the data are realizations of a random process. These are statistics derived methods, sequential nonparametric estimation algorithms in particular. There are a number of such algorithms such as those based on stochastic approximation or
Hermite series estimators.
2198:
Computing approximate quantiles from data arriving from a stream can be done efficiently using compressed data structures. The most popular methods are t-digest and KLL. These methods read a stream of values in a continuous fashion and can, at any time, be queried about the approximate value of a
1772:
The 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. Moreover, the sample median is a
531:
The rank of the second quartile (same as the median) is 10×(2/4) = 5, which is an integer, while the number of values (10) is an even number, so the average of both the fifth and sixth values is taken—that is (8+10)/2 = 9, though any value from 8 through to 10 could be taken to be the
2230:
These 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. The statistics based algorithms do present certain advantages however, particularly in the
2226:
The 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. These algorithms are computer science derived methods. Another class of
603:
The first quartile is determined by 11×(1/4) = 2.75, which rounds up to 3, meaning that 3 is the rank in the population (from least to greatest values) at which approximately 1/4 of the values are less than the value of the first quartile. The third value in the population is 7.
519:
The rank of the first quartile is 10×(1/4) = 2.5, which rounds up to 3, meaning that 3 is the rank in the population (from least to greatest values) at which approximately 1/4 of the values are less than the value of the first quartile. The third value in the population is 7.
2010:
757:
1875:
2222:
with useful properties: t-digest or KLL sketches can be combined. Computing 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.
961:
1105:
2207:-means clustering to group similar values. The 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
2251:, any particular sample of this random variable will have roughly a 63% chance of being less than the mean. This is because the exponential distribution has a long tail for positive values but is zero for negative numbers.
2289:, unless an arbitrary choice has been made from a range of values to specify a particular quantile. (See quantile estimation, above, for examples of such interpolation.) Quantiles can also be used in cases where only
411:
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
2243:
between (in this case) the 80th and the 81st scalar percentile. This separate meaning of percentile is also used in peer-reviewed scientific research articles. The meaning used can be derived from its context.
2184:
659:
1886:
1776:
One 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,
2247:
If a distribution is symmetric, then the median is the mean (so long as the latter exists). But, in general, the median and the mean can differ. For instance, with a random variable that has an
135:(100 groups). The 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.
2190:
Ma, Y., Genton, M. G., & Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. Annals of the
Institute of Statistical Mathematics, 63(2), 227–243.
861:
1782:
1012:
1261:
The 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
2231:
non-stationary streaming setting i.e. time-varying data. The algorithms of both classes, along with some respective advantages and disadvantages have been recently surveyed.
2083:
468:) regard the minimum and maximum as the 0th and 100th percentile, respectively. However, this broader terminology is an extension beyond traditional statistics definitions.
2239:
Standardized test results are commonly reported as a student scoring "in the 80th percentile", for example. This uses an alternative meaning of the word percentile as the
592:
Although not universally accepted, one can also speak of the zeroth quartile. This is the minimum value of the set, so the zeroth quartile in this example would be 3.
508:
Although not universally accepted, one can also speak of the zeroth quartile. This is the minimum value of the set, so the zeroth quartile in this example would be 3.
656:
For any population probability distribution on finitely many values, and generally for any probability distribution with a mean and variance, it is the case that
2095:
A 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
648:
So the first, second and third 4-quantiles (the "quartiles") of the dataset are . If also required, the zeroth quartile is 3 and the fourth quartile is 20.
627:
The third quartile value for the original example above is determined by 11×(3/4) = 8.25, which rounds up to 9. The ninth value in the population is 15.
564:
So the first, second and third 4-quantiles (the "quartiles") of the dataset are . If also required, the zeroth quartile is 3 and the fourth quartile is 20.
1554:
Linear interpolation of the expectations for the order statistics for the uniform distribution on . That is, it is the linear interpolation between points
2523:
3067:
2088:
But when the distribution is discrete, then the distribution of the sample median and the other quantiles fails to be Normal (see examples in
2730:
1722:
Excel's PERCENTILE and PERCENTILE.INC and Python's optional "inclusive" method are equivalent to R‑7. This is R's and Julia's default method.
2101:
1143:
One problem which frequently arises is estimating a quantile of a (very large or infinite) population based on a finite sample of size
1315:
includes two. Mathematica, SciPy and Julia support arbitrary parameters for methods which allow for other, non-standard, methods.
476:
The following two examples use the
Nearest Rank definition of quantile with rounding. For an explanation of this definition, see
2005:{\displaystyle {\text{Sample quantile p}}\sim {\mathcal {N}}\left(\mu =x_{p},\sigma ^{2}={\frac {p(1-p)}{Nf(x_{p})^{2}}}\right)}
1753:
Of the techniques, Hyndman and Fan recommend R-8, but most statistical software packages have chosen R-6 or R-7 as the default.
2266:
shares the ability to be relatively insensitive to large deviations in outlying observations, although even better methods of
3171:
2844:
2657:
117:
in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as
2694:
1534:
Piecewise linear function where the knots are the values midway through the steps of the empirical distribution function.
461:
2255:
from the mean, then quantiles may be more useful descriptive statistics than means and other moment-related statistics.
1224:, is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from
3107:"Unskilled and unaware of it: how difficulties in recognizing one's own incompetence lead to inflated self-assessments"
543:
The rank of the third quartile is 10×(3/4) = 7.5, which rounds up to 8. The eighth value in the population is 15.
396:
is not an integer, then round up to the next integer to get the appropriate index; the corresponding data value is the
2273:
The quantiles of a random variable are preserved under increasing transformations, in the sense that, for example, if
3239:
752:{\displaystyle \mu -\sigma \cdot {\sqrt {\frac {1-p}{p}}}\leq Q(p)\leq \mu +\sigma \cdot {\sqrt {\frac {p}{1-p}}}\,,}
2482:
230:
208:
2712:
2624:
1870:{\displaystyle {\text{Sample median m}}\sim {\mathcal {N}}\left(\mu =m,\sigma ^{2}={\frac {1}{Nf(m)^{2}}}\right)}
1368:
2869:
572:
Consider an ordered population of 11 data values . What are the 4-quantiles (the "quartiles") of this dataset?
488:
Consider an ordered population of 10 data values . What are the 4-quantiles (the "quartiles") of this dataset?
3081:
1308:
2611:
195:(2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to
2734:
1300:
2756:
2546:
1749:. Choices include returning an error value, computing linear extrapolation, or assuming a constant value.
3277:
2789:
1757:
1255:
2647:
1725:
Packages differ in how they estimate quantiles beyond the lowest and highest values in the sample, i.e.
3261:
3026:
Stephanou, M. and
Varughese, M (2023). "Hermiter: R package for sequential nonparametric estimation".
2885:
Dunning, Ted; Ertl, Otmar (February 2019). "Computing
Extremely Accurate Quantiles Using t-Digests".
2811:
2263:
2259:
2036:
191:. In some cases the value of a quantile may not be uniquely determined, as can be the case for the
2964:
Proceedings of the sixth ACM SIGKDD international conference on
Knowledge discovery and data mining
2248:
1276:
956:{\displaystyle \mu +z\sigma \geq Q\left({\frac {z^{2}}{1+z^{2}}}\right)\,,\mathrm {~for~} z\geq 0.}
110:
106:
2424:
The 1000-quantiles have been called permilles or milliles, but these are rare and largely obsolete
1674:
The resulting quantile estimates are approximately unbiased for the expected order statistics if
272:
drawn from it. For a population, of discrete values or for a continuous population density, the
3161:
2472:
2446:
1761:
1100:{\displaystyle \mu +z\sigma \leq Q\left({\frac {1}{1+z^{2}}}\right)\,,\mathrm {~for~} z\leq 0.}
265:
2673:
2454:– related to expectations in a way analogous to that in which quantiles are related to medians
3061:
1632:
Linear interpolation of the modes for the order statistics for the uniform distribution on .
2929:"A space-efficient recursive procedure for estimating a quantile of an unknown distribution"
2906:
Zohar Karnin; Kevin Lang; Edo
Liberty (2016). "Optimal Quantile Approximation in Streams".
2496:
8:
2477:
2441:
2327:
2219:
269:
196:
114:
24:
2666:
2301:
Values that divide sorted data into equal subsets other than four have different names.
3228:
3204:
3142:
3049:
3031:
2998:
2907:
2886:
2594:
2492:
2487:
1161:
810:
261:
148:
102:
1457:
3235:
3167:
3134:
3126:
3053:
2865:
2840:
2653:
2574:
2538:
2467:
2267:
1719:
Excel's PERCENTILE.EXC and Python's default "exclusive" method are equivalent to R‑6.
1151:
221:
1265:
used to choose the point along the piecewise linear interpolation curve, is chosen.
460:
in the above formulas. This broader terminology is used when quantiles are used to
3196:
3146:
3118:
3041:
3008:
2967:
2940:
2586:
1691:
R‑4 and following are piecewise linear, without discontinuities, but differ in how
1292:
226:
27:, with quantiles shown. The area below the red curve is the same in the intervals
1312:
802:
465:
212:
204:
200:
3122:
2790:"statistics — Mathematical statistics functions — Python 3.8.3rc1 documentation"
2400:
2089:
3045:
2807:
2643:
2570:
1284:
1157:
850:
280:-quantile is the data value where the cumulative distribution function crosses
2411:
3271:
3130:
2542:
2290:
3106:
2776:
2358:
1518:
Linear interpolation of the inverse of the empirical distribution function.
1164:
of nine algorithms used by various software packages. All methods compute
264:, the estimation of a quantile depends upon whether one is operating with a
3138:
2462:
1539:
R‑6, Excel, Python, SAS‑4, SciPy‑(0,0), Julia-(0,0), Maple‑5, Stata‑altdef
371:-quantile of this population can equivalently be computed via the value of
16:
Statistical method of dividing data into equal-sized intervals for analysis
2971:
2407:
2376:
2369:
2362:
2316:
2312:
2326:→ Q; the difference between upper and lower quartiles is also called the
1268:
432:
94:
2959:
2383:
3208:
3012:
2598:
2418:
1280:
477:
152:
131:
90:
1653:
Linear interpolation of the approximate medians for order statistics.
988:
standard deviations above the mean is always greater than or equal to
2986:
2451:
2435:
970:
standard deviation above the mean is always greater than or equal to
3200:
2944:
2590:
109:
into continuous intervals with equal probabilities, or dividing the
19:
3036:
3003:
2912:
2891:
2457:
2323:
1767:
119:
2928:
2905:
2179:{\displaystyle F_{\text{mid}}(x)=P(X\leq x)-{\frac {1}{2}}P(X=x)}
174:
1283:
programming languages support all nine sample quantile methods.
3256:
3187:
Stephen B. Vardeman (1992). "What about the Other
Intervals?".
2985:
Stephanou, Michael; Varughese, Melvin; Macdonald, Iain (2017).
2390:
2306:
1296:
1272:
192:
159:
125:
3025:
2695:"scipy.stats.mstats.mquantiles — SciPy v1.4.1 Reference Guide"
1150:
Modern statistical packages rely on a number of techniques to
1688:
R‑1 through R‑3 are piecewise constant, with discontinuities.
1616:
R‑7, Excel, Python, SciPy‑(1,1), Julia-(1,1), Maple‑6, NumPy
1304:
1288:
2987:"Sequential quantiles via Hermite series density estimation"
2984:
2674:"Function Reference: quantile – Octave-Forge – SourceForge"
1760:
of a quantile estimate can in general be estimated via the
1318:
The estimate types and interpolation schemes used include:
2396:
The 12-quantiles are called duo-deciles or dodeciles → DD
829:
is never more than one standard deviation from the mean.
2837:
Introduction to Robust Estimation and Hypothesis Testing
2524:"Glimpses of inequalities in probability and statistics"
1414:
The same as R-1, but with averaging at discontinuities.
2884:
1698:
R‑3 and R‑4 are not symmetric in that they do not give
413:
2960:"Incremental quantile estimation for massive tracking"
2193:
3019:
2777:
Stata documentation for the pctile and xtile commands
2104:
2039:
1889:
1785:
1138:
1015:
864:
662:
2933:
SIAM Journal on Scientific and Statistical Computing
3186:
2957:
3227:
2958:Chen, Fei; Lambert, Diane; Pinheiro, Jose (2000).
2178:
2077:
2004:
1869:
1099:
955:
751:
3230:Approximation Theorems of Mathematical Statistics
832:The above formula can be used to bound the value
462:parameterize continuous probability distributions
3269:
2585:(4). American Statistical Association: 361–365.
2029:is the value of the distribution density at the
1768:The asymptotic distribution of the sample median
1764:. The Maritz–Jarrett method can also be used.
1658:R‑9, SciPy‑(3/8,3/8), Julia‑(3/8,3/8), Maple‑8
1637:R‑8, SciPy‑(1/3,1/3), Julia‑(1/3,1/3), Maple‑7
1523:R‑5, SciPy‑(1/2,1/2), Julia‑(1/2,1/2), Maple‑4
464:. Moreover, some software programs (including
1465:R‑4, SAS‑1, SciPy‑(0,1), Julia‑(0,1), Maple‑3
1458:choosing the even integer in the case of a tie
3104:
2978:
2641:
2531:International Journal of Statistical Sciences
2090:https://stats.stackexchange.com/a/86638/28746
199:distributions, providing a way to generalize
3111:Journal of Personality and Social Psychology
3066:: CS1 maint: multiple names: authors list (
2521:
1773:more robust estimator than the sample mean.
1603:) randomly drawn values will not exceed the
1303:include the six piecewise linear functions,
255:
2920:
2878:
1456:indicates rounding to the nearest integer,
651:
2899:
2853:
2575:"Sample Quantiles in Statistical Packages"
2569:
3159:
3105:Kruger, J.; Dunning, D. (December 1999).
3035:
3002:
2911:
2890:
2859:
1067:
923:
745:
3225:
3180:
3160:Walker, Helen Mary; Lev, Joseph (1969).
2951:
2296:
483:
18:
2926:
2862:Kendall's Advanced Theory of Statistics
2806:
1287:includes five sample quantile methods,
567:
260:As in the computation of, for example,
3270:
3153:
2834:
2713:"Statistics – Maple Programming Help"
2438:– sort by first bucketing by quantile
2214:Both methods belong to the family of
1880:This extends to the other quantiles,
1596:is the probability that the last of (
219:-quantiles are the application of the
1444:The observation numbered closest to
981:, the median, and the value that is
162:of (nearly) equal sizes. There are
2779:See 'Methods and formulas' section.
2277:is the median of a random variable
2194:Approximate quantiles from a stream
13:
3219:
2757:"Statistics – Julia Documentation"
2305:The only 2-quantile is called the
2258:Closely related is the subject of
1900:
1796:
1139:Estimating quantiles from a sample
1081:
1078:
1075:
1009:, there is instead an upper bound
937:
934:
931:
856:standard deviations above the mean
414:Estimating quantiles from a sample
404:-quantile. On the other hand, if
14:
3289:
3249:
2860:Stuart, Alan; Ord, Keith (1994).
2812:"Sample quantiles 20 years later"
1201:by computing a real valued index
3255:
2991:Electronic Journal of Statistics
356:equally probable values indexed
231:cumulative distribution function
209:cumulative distribution function
3098:
3074:
2828:
2800:
2782:
2770:
2749:
2522:Bagui, S.; Bhaumik, D. (2004).
2078:{\displaystyle x_{p}=F^{-1}(p)}
1369:empirical distribution function
963:For example, the value that is
3166:. Holt, Rinehart and Winston.
3163:Elementary Statistical Methods
2723:
2705:
2687:
2635:
2617:
2605:
2573:; Fan, Yanan (November 1996).
2563:
2515:
2173:
2161:
2142:
2130:
2121:
2115:
2072:
2066:
1985:
1971:
1960:
1948:
1850:
1843:
707:
701:
419:If, instead of using integers
1:
2508:
2417:The 100-quantiles are called
2234:
813:. In particular, the median
639:fourth quartile would be 11.
555:fourth quartile would be 10.
203:to continuous variables (see
2406:The 20-quantiles are called
2399:The 16-quantiles are called
2389:The 10-quantiles are called
842:in terms of quantiles. When
363:from lowest to highest, the
101:are cut points dividing the
7:
3123:10.1037/0022-3514.77.6.1121
2428:
2382:The 8-quantiles are called
2375:The 7-quantiles are called
2368:The 6-quantiles are called
2357:The 5-quantiles are called
2322:The 4-quantiles are called
2311:The 3-quantiles are called
1376:R‑2, SAS‑5, Maple‑2, Stata
1256:floor and ceiling functions
471:
352:For a finite population of
10:
3296:
3046:10.1007/s00180-023-01382-0
1607:-th smallest of the first
3234:. John Wiley & Sons.
3189:The American Statistician
2612:Mathematica Documentation
2264:Least absolute deviations
2260:least absolute deviations
2033:-th population quantile (
1678:is normally distributed.
1472:
775:(or equivalently is the
431:-quantile" is based on a
302:-quantile for a variable
256:Quantiles of a population
173:-quantiles, one for each
23:Probability density of a
3226:Serfling, R. J. (1980).
3028:Computational Statistics
2835:Wilcox, Rand R. (2010).
2249:exponential distribution
1197:) from a sample of size
652:Relationship to the mean
107:probability distribution
1611:randomly drawn values.
1171:, the estimate for the
1107:For example, the value
999:, the fourth quintile.
2927:Tierney, Luke (1983).
2625:"Quantile calculation"
2473:Quantile normalization
2447:Descriptive statistics
2180:
2079:
2006:
1871:
1254:. (For notation, see
1101:
957:
809:is the distribution's
801:is the distribution's
753:
266:statistical population
86:
2972:10.1145/347090.347195
2614:See 'Details' section
2579:American Statistician
2414:, or demi-deciles → V
2297:Other quantifications
2181:
2080:
2007:
1872:
1102:
958:
754:
616:the population is 9.
484:Even-sized population
22:
3264:at Wikimedia Commons
2497:confidence intervals
2293:data are available.
2220:Streaming Algorithms
2218:that are subsets of
2199:specified quantile.
2102:
2037:
1887:
1783:
1345:R‑1, SAS‑3, Maple‑1
1295:both include eight,
1213:-th smallest of the
1135:, the first decile.
1013:
862:
849:, the value that is
763:is the value of the
660:
568:Odd-sized population
2478:Quantile regression
2442:Interquartile range
2328:interquartile range
1209:is an integer, the
1160:and Fan compiled a
25:normal distribution
3278:Summary statistics
3013:10.1214/17-EJS1245
2864:. London: Arnold.
2839:. Academic Press.
2493:Tolerance interval
2488:Summary statistics
2176:
2075:
2002:
1867:
1311:includes two, and
1124:will never exceed
1097:
953:
858:has a lower bound
811:standard deviation
749:
320:or, equivalently,
262:standard deviation
169:partitions of the
129:(ten groups), and
87:
3260:Media related to
3173:978-0-03-081130-2
2846:978-0-12-751542-7
2810:(28 March 2016).
2737:on April 16, 2016
2717:www.maplesoft.com
2659:978-3-900051-07-5
2468:Quantile function
2285:is the median of
2268:robust regression
2156:
2112:
1995:
1893:
1892:Sample quantile p
1860:
1789:
1682:
1681:
1183:-quantile, where
1086:
1074:
1061:
942:
930:
917:
743:
742:
693:
692:
646:
645:
562:
561:
222:quantile function
3285:
3259:
3245:
3233:
3213:
3212:
3184:
3178:
3177:
3157:
3151:
3150:
3117:(6): 1121–1134.
3102:
3096:
3095:
3093:
3092:
3086:Oxford Reference
3078:
3072:
3071:
3065:
3057:
3039:
3023:
3017:
3016:
3006:
2982:
2976:
2975:
2955:
2949:
2948:
2924:
2918:
2917:
2915:
2903:
2897:
2896:
2894:
2882:
2876:
2875:
2857:
2851:
2850:
2832:
2826:
2825:
2823:
2822:
2804:
2798:
2797:
2786:
2780:
2774:
2768:
2767:
2765:
2763:
2753:
2747:
2746:
2744:
2742:
2733:. Archived from
2727:
2721:
2720:
2709:
2703:
2702:
2691:
2685:
2684:
2682:
2680:
2670:
2664:
2663:
2649:Sample Quantiles
2639:
2633:
2632:
2629:uk.mathworks.com
2621:
2615:
2609:
2603:
2602:
2567:
2561:
2560:
2558:
2557:
2551:
2545:. Archived from
2528:
2519:
2353:
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2284:
2280:
2276:
2210:
2185:
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2182:
2177:
2157:
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2110:
2084:
2082:
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2008:
2003:
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1994:
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1992:
1983:
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1943:
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1937:
1925:
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1515:
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1182:
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1170:
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1106:
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1103:
1098:
1087:
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1062:
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1039:
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962:
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922:
918:
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915:
914:
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888:
855:
848:
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828:
808:
800:
796:
782:
778:
774:
766:
762:
758:
756:
755:
750:
744:
741:
727:
726:
694:
688:
677:
676:
635:Fourth quartile
612:Second quartile
589:Zeroth quartile
575:
574:
551:Fourth quartile
528:Second quartile
505:Zeroth quartile
491:
490:
459:
449:
445:
437:
430:
426:
422:
410:
403:
399:
395:
388:
370:
366:
362:
355:
347:
330:
319:
305:
301:
297:
293:
289:
279:
275:
251:
233:) to the values
227:inverse function
218:
190:
179:
172:
168:
158:
147:are values that
141:
84:
73:
55:
37:
3295:
3294:
3288:
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3286:
3284:
3283:
3282:
3268:
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3252:
3242:
3222:
3220:Further reading
3217:
3216:
3201:10.2307/2685212
3185:
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2883:
2879:
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2854:
2847:
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2820:
2818:
2808:Hyndman, Rob J.
2805:
2801:
2794:docs.python.org
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2787:
2783:
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2759:
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2731:"EViews 9 Help"
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2644:Hyndman, Rob J.
2640:
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2623:
2622:
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2610:
2606:
2591:10.2307/2684934
2571:Hyndman, Rob J.
2568:
2564:
2555:
2553:
2549:
2526:
2520:
2516:
2511:
2506:
2431:
2421:or centiles → P
2352:
2345:
2339:
2299:
2286:
2282:
2278:
2274:
2270:are available.
2237:
2208:
2196:
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2100:
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2016:
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1788:Sample median m
1786:
1784:
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1770:
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1726:
1710:
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1692:
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1661:
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1313:Microsoft Excel
1262:
1252:
1242:
1239:
1229:
1225:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1184:
1180:
1176:
1175:-quantile (the
1172:
1169:
1165:
1154:the quantiles.
1144:
1141:
1125:
1118:
1108:
1071:
1054:
1050:
1043:
1038:
1034:
1014:
1011:
1010:
1003:
989:
982:
971:
964:
927:
910:
906:
899:
893:
889:
887:
883:
863:
860:
859:
851:
843:
833:
814:
806:
803:arithmetic mean
798:
784:
780:
776:
768:
764:
760:
731:
725:
678:
675:
661:
658:
657:
654:
624:Third quartile
600:First quartile
570:
540:Third quartile
516:First quartile
486:
474:
466:Microsoft Excel
451:
447:
439:
435:
428:
424:
420:
409:
405:
401:
397:
394:
390:
376:
372:
368:
364:
357:
353:
338:
321:
310:
303:
299:
295:
291:
281:
277:
273:
258:
234:
216:
213:random variable
205:percentile rank
201:rank statistics
181:
177:
170:
163:
156:
155:of values into
139:
123:(four groups),
82:
75:
71:
64:
57:
53:
46:
39:
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28:
17:
12:
11:
5:
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3251:
3250:External links
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3247:
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3218:
3215:
3214:
3195:(3): 193–197.
3179:
3172:
3152:
3097:
3073:
3018:
2997:(1): 570-607.
2977:
2950:
2939:(4): 706-711.
2919:
2898:
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2852:
2845:
2827:
2816:Hyndsignt blog
2799:
2781:
2769:
2748:
2722:
2704:
2699:docs.scipy.org
2686:
2665:
2658:
2642:Frohne, Ivan;
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1825:
1821:
1817:
1814:
1811:
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1804:
1798:
1793:
1769:
1766:
1758:standard error
1751:
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1723:
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783:-quantile for
767:-quantile for
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331:
257:
254:
215:is known, the
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51:
44:
33:
15:
9:
6:
4:
3:
2:
3291:
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3241:0-471-02403-1
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2708:
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2652:. R Project.
2651:
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2613:
2608:
2600:
2596:
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2588:
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2576:
2572:
2566:
2552:on 2021-08-12
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2536:
2532:
2525:
2518:
2514:
2503:th quantile")
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2216:data sketches
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1989:
1979:
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1968:
1965:
1957:
1954:
1951:
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1921:
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26:
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3110:
3100:
3089:. Retrieved
3085:
3082:"percentile"
3076:
3062:cite journal
3027:
3021:
2994:
2990:
2980:
2963:
2953:
2936:
2932:
2922:
2901:
2880:
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2855:
2836:
2830:
2819:. Retrieved
2815:
2802:
2793:
2784:
2772:
2760:. Retrieved
2751:
2739:. Retrieved
2735:the original
2725:
2716:
2707:
2698:
2689:
2677:. Retrieved
2668:
2648:
2637:
2628:
2619:
2607:
2582:
2578:
2565:
2554:. Retrieved
2547:the original
2534:
2530:
2517:
2500:
2483:Quantization
2347:
2340:
2336:middle fifty
2335:
2331:
2300:
2272:
2257:
2253:
2246:
2240:
2238:
2229:
2225:
2215:
2213:
2204:
2201:
2197:
2188:
2094:
2087:
2021:
2017:
2014:
1879:
1775:
1771:
1755:
1752:
1745:
1741:
1737:
1731:
1727:
1711:
1704:
1700:
1695:is computed.
1683:
1667:
1663:
1646:
1642:
1625:
1621:
1598:
1591:
1587:
1582:
1578:
1570:
1566:
1561:
1557:
1548:
1544:
1527:
1509:
1504:
1498:
1493:
1489:
1485:
1479:
1474:
1451:
1436:
1431:
1423:
1405:
1400:
1394:
1389:
1380:
1359:
1354:
1317:
1267:
1260:
1248:
1243:
1235:
1230:
1193:
1189:
1185:
1156:
1149:
1142:
1130:
1126:
1119:
1113:
1109:
1004:
1001:
994:
990:
983:
976:
972:
965:
852:
844:
838:
834:
831:
824:
820:
816:
805:, and where
793:
789:
785:
770:
655:
647:
581:Calculation
571:
563:
497:Calculation
487:
475:
456:
452:
441:
418:
385:
381:
378:
359:
351:
344:
340:
333:
327:
323:
316:
312:
290:. That is,
286:
282:
259:
248:
244:
240:
236:
220:
187:
183:
164:
144:
138:
137:
130:
124:
118:
111:observations
98:
88:
77:
66:
59:
48:
41:
30:
2966:: 516-522.
2679:6 September
2419:percentiles
2401:hexadeciles
1419:R‑3, SAS‑2
1367:Inverse of
1269:Mathematica
478:percentiles
433:real number
180:satisfying
132:percentiles
95:probability
3091:2020-08-17
3037:2111.14091
3004:1507.05073
2913:1603.05346
2892:1902.04023
2871:0340614307
2821:2020-11-30
2556:2021-08-12
2509:References
2412:vigintiles
2235:Discussion
1281:GNU Octave
268:or with a
197:continuous
153:finite set
91:statistics
3262:Quantiles
3131:0022-3514
3054:244715035
2543:1683-5603
2452:Expectile
2436:Flashsort
2359:quintiles
2332:midspread
2324:quartiles
2146:−
2137:≤
2059:−
1955:−
1931:σ
1911:μ
1896:∼
1820:σ
1807:μ
1792:∼
1762:bootstrap
1448:. Here,
1092:≤
1029:≤
1026:σ
1017:μ
948:≥
878:≥
875:σ
866:μ
797:), where
736:−
723:⋅
720:σ
714:μ
711:≤
696:≤
683:−
673:⋅
670:σ
667:−
664:μ
578:Quartile
494:Quartile
450:replaces
322:Pr ≥ 1 −
149:partition
145:quantiles
120:quartiles
99:quantiles
3272:Category
3139:10626367
2762:June 17,
2741:April 4,
2646:(2009).
2537:: 9–15.
2499:for the
2463:Q–Q plot
2458:Quartile
2429:See also
2408:ventiles
2377:septiles
2370:sextiles
2363:pentiles
2338:→ IQR =
2317:terciles
2313:tertiles
2241:interval
1707:+ 1) / 2
1576:, where
1217:values,
1205:. When
1162:taxonomy
1152:estimate
532:median.
472:Examples
3209:2685212
3147:2109278
2599:2684934
2391:deciles
2384:octiles
2291:ordinal
2281:, then
1730:< 1/
1684:Notes:
1158:Hyndman
769:0 <
584:Result
500:Result
440:0 <
427:, the "
229:of the
182:0 <
175:integer
160:subsets
126:deciles
3238:
3207:
3170:
3145:
3137:
3129:
3052:
2868:
2843:
2656:
2597:
2541:
2307:median
2015:where
1740:> (
1666:+ 1/4)
1645:+ 1/3)
1408:+ 1/2⌋
1397:– 1/2⌉
1340:Notes
1309:Python
1297:EViews
1273:Matlab
1241:, and
1133:= 0.1)
1085:
1073:
997:= 0.8)
979:= 0.5)
941:
929:
827:= 1/2)
773:< 1
759:where
444:< 1
389:. If
358:1, …,
270:sample
243:, …, (
193:median
115:sample
74:, and
3205:JSTOR
3143:S2CID
3050:S2CID
3032:arXiv
2999:arXiv
2908:arXiv
2887:arXiv
2595:JSTOR
2550:(PDF)
2527:(PDF)
2085:).
1744:− 1)/
1714:= 1/2
1709:when
1670:+ 3/8
1649:+ 1/3
1530:+ 1/2
1426:− 1/2
1410:) / 2
1383:+ 1/2
1324:Type
1305:Stata
1301:Julia
1293:Maple
1289:SciPy
1002:When
446:then
438:with
339:Pr ≥
311:Pr ≤
294:is a
247:− 1)/
225:(the
211:of a
186:<
113:in a
105:of a
103:range
3236:ISBN
3168:ISBN
3135:PMID
3127:ISSN
3068:link
2866:ISBN
2841:ISBN
2764:2023
2743:2016
2681:2013
2654:ISBN
2539:ISSN
2379:→ SP
2365:→ QU
1756:The
1735:and
1624:− 1)
1547:+ 1)
1492:⌋) (
1299:and
1291:and
1279:and
1179:-th
1122:= −3
1117:for
779:-th
761:Q(p)
423:and
400:-th
367:-th
334:and
298:-th
276:-th
239:, 2/
93:and
83:,+∞)
29:(−∞,
3197:doi
3119:doi
3042:doi
3009:doi
2968:doi
2941:doi
2587:doi
2403:→ H
2393:→ D
2386:→ O
2372:→ S
2361:or
2334:or
2319:→ T
2315:or
2111:mid
2092:).
1703:= (
1628:+ 1
1594:+1)
1488:− ⌊
1484:+ (
1285:SAS
1258:).
1007:≤ 0
986:= 2
968:= 1
847:≥ 0
642:20
630:15
558:20
546:15
416:).
306:if
252:}.
235:{1/
167:− 1
89:In
3274::
3203:.
3193:46
3191:.
3141:.
3133:.
3125:.
3115:77
3113:.
3109:.
3084:.
3064:}}
3060:{{
3048:.
3040:.
3030:.
3007:.
2995:11
2993:.
2989:.
2962:.
2935:.
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2814:.
2792:.
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2697:.
2627:.
2593:.
2583:50
2581:.
2577:.
2533:.
2529:.
2495:("
2410:,
2346:−
2330:,
2211:.
1601:+1
1590:/(
1586:=
1565:,
1528:Np
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1469:Np
1460:.
1446:Np
1424:Np
1399:+
1381:Np
1371:.
1349:Np
1275:,
1271:,
1228:,
1188:=
1147:.
1114:zσ
1112:+
1095:0.
951:0.
839:zσ
837:+
819:=
788:=
619:9
607:7
595:3
535:9
523:7
511:3
480:.
377:=
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3244:.
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2165:X
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2131:(
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2125:=
2122:)
2119:x
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2107:F
2073:)
2070:p
2067:(
2062:1
2055:F
2051:=
2046:p
2042:x
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2027:)
2024:p
2022:x
2020:(
2018:f
1999:)
1990:2
1986:)
1980:p
1976:x
1972:(
1969:f
1966:N
1961:)
1958:p
1952:1
1949:(
1946:p
1940:=
1935:2
1927:,
1922:p
1918:x
1914:=
1907:(
1901:N
1864:)
1855:2
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1844:(
1841:f
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1834:1
1829:=
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1803:(
1797:N
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1738:p
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1728:p
1716:.
1712:p
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1701:h
1693:h
1676:x
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1571:h
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1556:(
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1543:(
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1512:⌋
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1501:⌉
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1497:⌈
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1388:(
1362:⌉
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1234:⌊
1231:x
1226:h
1221:h
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1207:h
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1199:N
1194:q
1192:/
1190:k
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1181:q
1177:k
1173:p
1168:p
1166:Q
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1131:p
1129:(
1127:Q
1120:z
1110:μ
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1064:)
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1052:z
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1032:Q
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993:(
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