161:
29:
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or simply through natural deviations in populations. A sample may have been contaminated with elements from outside the population being examined. Alternatively, an outlier could be the result of a flaw in the assumed theory, calling for further investigation by the researcher. Additionally, the pathological appearance of outliers of a certain form appears in a variety of datasets, indicating that the causative mechanism for the data might differ at the extreme end (
1818:
1689:
rejection, they do not make the practice more scientifically or methodologically sound, especially in small sets or where a normal distribution cannot be assumed. Rejection of outliers is more acceptable in areas of practice where the underlying model of the process being measured and the usual distribution of measurement error are confidently known.
1738:, the sample variance increases with the sample size, the sample mean fails to converge as the sample size increases, and outliers are expected at far larger rates than for a normal distribution. Even a slight difference in the fatness of the tails can make a large difference in the expected number of extreme values.
1323:
2013:, p. 1 stating "An outlying observation may be merely an extreme manifestation of the random variability inherent in the data. ... On the other hand, an outlying observation may be the result of gross deviation from prescribed experimental procedure or an error in calculating or recording the numerical value."
107:, or it may be that some observations are far from the center of the data. Outlier points can therefore indicate faulty data, erroneous procedures, or areas where a certain theory might not be valid. However, in large samples, a small number of outliers is to be expected (and not due to any anomalous condition).
1684:
When deciding whether to remove an outlier, the cause has to be considered. As mentioned earlier, if the outlier's origin can be attributed to an experimental error, or if it can be otherwise determined that the outlying data point is erroneous, it is generally recommended to remove it. However, it
699:
The modified
Thompson Tau test is a method used to determine if an outlier exists in a data set. The strength of this method lies in the fact that it takes into account a data set's standard deviation, average and provides a statistically determined rejection zone; thus providing an objective method
180:
or more from the mean, and 1 in 370 will deviate by three times the standard deviation. In a sample of 1000 observations, the presence of up to five observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice
1675:
Even when a normal distribution model is appropriate to the data being analyzed, outliers are expected for large sample sizes and should not automatically be discarded if that is the case. Instead, one should use a method that is robust to outliers to model or analyze data with naturally occurring
228:
Outliers can have many anomalous causes. A physical apparatus for taking measurements may have suffered a transient malfunction. There may have been an error in data transmission or transcription. Outliers arise due to changes in system behaviour, fraudulent behaviour, human error, instrument error
133:
of the data will be between 20 and 25 °C but the mean temperature will be between 35.5 and 40 °C. In this case, the median better reflects the temperature of a randomly sampled object (but not the temperature in the room) than the mean; naively interpreting the mean as "a typical sample",
241:
There is no rigid mathematical definition of what constitutes an outlier; determining whether or not an observation is an outlier is ultimately a subjective exercise. There are various methods of outlier detection, some of which are treated as synonymous with novelty detection. Some are graphical
1688:
Removing a data point solely because it is an outlier, on the other hand, is a controversial practice, often frowned upon by many scientists and science instructors, as it typically invalidates statistical results. While mathematical criteria provide an objective and quantitative method for data
349:
such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the
825:
1700:. Trimming discards the outliers whereas Winsorising replaces the outliers with the nearest "nonsuspect" data. Exclusion can also be a consequence of the measurement process, such as when an experiment is not entirely capable of measuring such extreme values, resulting in
973:
is removed if it is an outlier). Meaning, if a data point is found to be an outlier, it is removed from the data set and the test is applied again with a new average and rejection region. This process is continued until no outliers remain in a data set.
700:
to determine if a data point is an outlier. How it works: First, a data set's average is determined. Next the absolute deviation between each data point and the average are determined. Thirdly, a rejection region is determined using the formula:
1557:
1043:
56:
that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the
977:
Some work has also examined outliers for nominal (or categorical) data. In the context of a set of examples (or instances) in a data set, instance hardness measures the probability that an instance will be misclassified (
253:
Model-based methods which are commonly used for identification assume that the data are from a normal distribution, and identify observations which are deemed "unlikely" based on mean and standard deviation:
706:
2708:
Bakker, Marjan; Wicherts, Jelte M. (2014). "Outlier removal, sum scores, and the inflation of the type I error rate in independent samples t tests: The power of alternatives and recommendations".
956:
1048:
118:, or both, depending on whether they are extremely high or low. However, the sample maximum and minimum are not always outliers because they may not be unusually far from other observations.
84:. A frequent cause of outliers is a mixture of two distributions, which may be two distinct sub-populations, or may indicate 'correct trial' versus 'measurement error'; this is modeled by a
868:
1602:
185:– and not indicate an anomaly. If the sample size is only 100, however, just three such outliers are already reason for concern, being more than 11 times the expected number.
1318:{\displaystyle {\begin{aligned}IH(\langle x,y\rangle )&=\sum _{H}(1-p(y,x,h))p(h|t)\\&=\sum _{H}p(h|t)-p(y,x,h)p(h|t)\\&=1-\sum _{H}p(y,x,h)p(h|t).\end{aligned}}}
1401:
1393:
1019:
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550:
1367:
1629:
603:
419:
392:
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problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the estimated coefficients, using a measure such as
220:
is approximately 0.3%, and thus for 1000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a
Poisson distribution with λ = 3.
573:
347:
327:
2090:
2800:
Cook, R. Dennis (Feb 1977). "Detection of
Influential Observations in Linear Regression". Technometrics (American Statistical Association) 19 (1): 15–18.
2590:(2012). "Local outlier detection reconsidered: A generalized view on locality with applications to spatial, video, and network outlier detection".
1850:
In cases where the cause of the outliers is known, it may be possible to incorporate this effect into the model structure, for example by using a
350:
probability of making so many, and no more, abnormal observations. (Quoted in the editorial note on page 516 to Peirce (1982 edition) from
329:
observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as
121:
Naive interpretation of statistics derived from data sets that include outliers may be misleading. For example, if one is calculating the
2152:
Pimentel, M. A., Clifton, D. A., Clifton, L., & Tarassenko, L. (2014). A review of novelty detection. Signal
Processing, 99, 215-249.
2001:
Pimentel, M. A., Clifton, D. A., Clifton, L., & Tarassenko, L. (2014). A review of novelty detection. Signal
Processing, 99, 215-249.
2863:
Roberts, S. and
Tarassenko, L.: 1995, A probabilistic resource allocating network for novelty detection. Neural Computation 6, 270–284.
2314:
2306:
2042:
820:{\displaystyle {\text{Rejection Region}}{=}{\frac {{t_{\alpha /2}}{\left(n-1\right)}}{{\sqrt {n}}{\sqrt {n-2+{t_{\alpha /2}^{2}}}}}}}
2953:
1949:
An outlying observation, or "outlier," is one that appears to deviate markedly from other members of the sample in which it occurs.
134:
equivalent to the median, is incorrect. As illustrated in this case, outliers may indicate data points that belong to a different
2166:
1663:
The choice of how to deal with an outlier should depend on the cause. Some estimators are highly sensitive to outliers, notably
2026:
2434:
2406:
2235:
2073:
1983:
885:
61:. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses.
1730:
The possibility should be considered that the underlying distribution of the data is not approximately normal, having "
64:
Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set,
2370:
Report of the
Superintendent of the United States Coast Survey Showing the Progress of the Survey During the Year 1870
2752:
2570:
1664:
2041:
Chandan
Mukherjee, Howard White, Marc Wuyts, 1998, "Econometrics and Data Analysis for Developing Countries Vol. 1"
1882:
834:
2129:
37:
2520:
2507:. Proceedings of the 2000 ACM SIGMOD international conference on Management of data - SIGMOD '00. p. 427.
2963:
2931:
882:
is the sample size, and s is the sample standard deviation. To determine if a value is an outlier: Calculate
263:
76:
to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high
1776:'s. When outliers occur, this intersection could be empty, and we should relax a small number of the sets
2926:
694:
2958:
1867:
1655:. Instance hardness provides a continuous value for determining if an instance is an outlier instance.
1565:
1552:{\displaystyle IH_{L}(\langle x,y\rangle )=1-{\frac {1}{|L|}}\sum _{j=1}^{|L|}p(y|x,g_{j}(t,\alpha ))}
2458:
Knorr, E. M.; Ng, R. T.; Tucakov, V. (2000). "Distance-based outliers: Algorithms and applications".
1033:). Ideally, instance hardness would be calculated by summing over the set of all possible hypotheses
69:
2625:
2391:(1986) . "On the Theory of Errors of Observation". In Kloesel, Christian J. W.; et al. (eds.).
2375:
2196:
2472:
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1851:
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is unknown for many algorithms. Thus, instance hardness can be approximated using a diverse subset
258:
193:
104:
20:
2937:
2091:"There and back again: Outlier detection between statistical reasoning and data mining algorithms"
196:
from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability
2769:
1897:
1892:
1783:(as small as possible) in order to avoid any inconsistency. This can be done using the notion of
1693:
243:
2638:
2628:".Journal of the Royal Statistical Society. Series B (Methodological), Vol. 47, No. 1, pp. 53-55
2398:
2392:
1372:
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2388:
2365:
2191:
1701:
135:
981:
2921:
1975:
1969:
1965:
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431:
303:
290:
are often used to detect outliers, especially in the development of linear regression models.
287:
201:
2057:
1931:
Grubbs, F. E. (February 1969). "Procedures for detecting outlying observations in samples".
1335:
1907:
1887:
1788:
1607:
678:
582:
397:
370:
283:
216:. Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean,
209:
182:
148:
capable of coping with outliers are said to be robust: the median is a robust statistic of
2823:
72:. In the case of measurement error, one wishes to discard them or use statistics that are
40:
displaying four outliers in the middle column, as well as one outlier in the first column.
8:
2541:
2257:(2012). "A survey on unsupervised outlier detection in high-dimensional numerical data".
2205:
2058:
1735:
1708:
677:
may take other approaches. Some of these may be distance-based and density-based such as
608:
364:
169:
139:
81:
425:
respectively, then one could define an outlier to be any observation outside the range:
2846:
2810:
2687:
2607:
2587:
2537:
2485:
2424:
2347:
2274:
2254:
2209:
2182:
Hodge, Victoria J.; Austin, Jim (2004), "A Survey of
Outlier Detection Methodologies",
2121:
1712:
969:
The modified
Thompson Tau test is used to find one outlier at a time (largest value of
682:
558:
332:
312:
181:
the expected number and hence within 1 standard deviation of the expected number – see
177:
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2725:
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2402:
2231:
2113:
2069:
1979:
1902:
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650:
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268:
73:
65:
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2626:
A Note on Restricted Maximum Likelihood Estimation with an Alternative Outlier Model
2611:
2489:
2125:
2023:
2882:
2842:
2838:
2781:
2717:
2671:
2599:
2558:
2508:
2477:
2397:. Vol. 3, 1872–1878. Bloomington, Indiana: Indiana University Press. pp.
2339:
2278:
2266:
2213:
2201:
2105:
1990:
An outlier is an observation that is far removed from the rest of the observations.
1944:
1940:
173:
160:
149:
96:
2291:
152:, while the mean is not. However, the mean is generally a more precise estimator.
2551:
Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data
2327:
2302:
2162:
2030:
666:
126:
92:
91:
In most larger samplings of data, some data points will be further away from the
2873:
Bishop, C. M. (August 1994). "Novelty detection and Neural Network validation".
80:
and that one should be very cautious in using tools or intuitions that assume a
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115:
111:
2886:
2786:
2603:
2947:
2905:
2117:
1961:
1855:
1719:
662:
293:
Subspace and correlation based techniques for high-dimensional numerical data
85:
125:
temperature of 10 objects in a room, and nine of them are between 20 and 25
2729:
2683:
2444:
658:
2562:
2512:
2481:
28:
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represent the input attribute value for an instance in the training set
2721:
2675:
646:
576:
53:
45:
2270:
2109:
2910:
2343:
1731:
145:
2368:(1873) . "Appendix No. 21. On the Theory of Errors of Observation".
2098:
Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery
2460:
The VLDB Journal the International Journal on Very Large Data Bases
1817:
1765:(instead of a probability density function). If no outliers occur,
422:
247:
77:
58:
33:
2064:. Studies in Computational Intelligence Vol. 5. Springer. p.
188:
In general, if the nature of the population distribution is known
2053:
654:
176:
means that roughly 1 in 22 observations will differ by twice the
122:
1685:
is more desirable to correct the erroneous value, if possible.
200:) of a given distribution, the number of outliers will follow a
19:
This article is about the statistical term. For other uses, see
16:
Observation far apart from others in statistics and data science
2554:
130:
110:
Outliers, being the most extreme observations, may include the
100:
2659:
2292:
E178: Standard Practice for Dealing With Outlying Observations
363:
Other methods flag observations based on measures such as the
280:
E178: Standard Practice for Dealing With Outlying Observations
95:
than what is deemed reasonable. This can be due to incidental
2824:"Probabilistic set-membership approach for robust regression"
2505:
Efficient algorithms for mining outliers from large data sets
2811:
Cauchy Distribution. From MathWorld--A Wolfram Web Resource
2745:
Data Analysis: A Statistical Primer for Psychology Students
277:
192:, it is possible to test if the number of outliers deviate
2413:– Appendix 21, according to the editorial note on page 515
1722:, this should be clearly stated on any subsequent report.
1814:-relaxed intersection could be suspected to be outliers.
2332:
Proceedings of the American Academy of Arts and Sciences
2056:; Chen, Guoqing; Kerre, Etienne (2005). Wets, G. (ed.).
962:> Rejection Region, the data point is an outlier. If
2637:
Smith, M.R.; Martinez, T.; Giraud-Carrier, C. (2014). "
2875:
IEE Proceedings - Vision, Image, and Signal Processing
2535:
2394:
Writings of Charles S. Peirce: A Chronological Edition
2307:"Criterion for the Rejection of Doubtful Observations"
966:≤ Rejection Region, the data point is not an outlier.
889:
838:
1641:
1610:
1568:
1404:
1375:
1338:
1046:
984:
888:
837:
709:
611:
585:
561:
434:
400:
373:
335:
315:
2770:"Simplified estimation from censored normal samples"
2585:
2502:
2252:
2060:
Intelligent Data Mining: Techniques and Applications
1795:-relaxed intersection corresponds to the set of all
1750:
considers that the uncertainty corresponding to the
951:{\displaystyle \scriptstyle \delta =|(X-mean(X))/s|}
685:to label observations as outliers or non-outliers.
681:(LOF). Some approaches may use the distance to the
2919:
1692:The two common approaches to exclude outliers are
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1623:
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208:, which can generally be well-approximated by the
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1718:If a data point (or points) is excluded from the
2945:
2902:
2330:(May 1877 – May 1878). "On Peirce's criterion".
1604:is the hypothesis induced by learning algorithm
645:In various domains such as, but not limited to,
1741:
1328:Practically, this formulation is unfeasible as
164:Relative probabilities in a normal distribution
2707:
2457:
2161:
2088:
1974:(2nd ed.). New York: MacMillan. pp.
2703:
2701:
2639:An Instance Level Analysis of Data Complexity
2543:LOF: Identifying Density-based Local Outliers
2503:Ramaswamy, S.; Rastogi, R.; Shim, K. (2000).
2052:
688:
537:
437:
2653:
2651:
2649:
2647:
2225:
1433:
1421:
1072:
1060:
863:{\displaystyle \scriptstyle {t_{\alpha /2}}}
236:
2664:Journal of Experimental Psychology: General
2660:"Outliers may not be automatically removed"
2326:
2248:
2246:
2181:
1754:th measurement of an unknown random vector
1725:
309:It is proposed to determine in a series of
2831:Journal of Statistical Theory and Practice
2747:. Transaction Publishers. pp. 24–25.
2698:
2785:
2644:
2471:
2195:
1769:should belong to the intersection of all
2243:
2089:Zimek, Arthur; Filzmoser, Peter (2018).
1816:
1658:
1332:is potentially infinite and calculating
695:Studentized residual § Distribution
159:
155:
27:
2168:Robust Regression and Outlier Detection
1960:
870:is the critical value from the Student
634:
2946:
2920:Balakrishnan, N.; Childs, A. (2001) ,
2872:
2821:
2387:
2378:(goes to Report p. 200, PDF's p. 215).
2364:
2010:
1930:
1734:". For instance, when sampling from a
2903:
2774:The Annals of Mathematical Statistics
2767:
2657:
2422:
2171:(3rd ed.), John Wiley & Sons
1845:
297:
129:, but an oven is at 175 °C, the
2742:
2641:". Machine Learning, 95(2): 225-256.
2259:Statistical Analysis and Data Mining
1825:-relaxed intersection of 6 sets for
1791:. As illustrated by the figure, the
103:that generated an assumed family of
2592:Data Mining and Knowledge Discovery
2226:Barnett, Vic; Lewis, Toby (1994) ,
13:
2206:10.1023/B:AIRE.0000045502.10941.a9
631:indicates data that is "far out".
14:
2975:
2896:
1665:estimation of covariance matrices
358:
2540:; Ng, R. T.; Sander, J. (2000).
1883:Data transformation (statistics)
1799:which belong to all sets except
1597:{\displaystyle g_{j}(t,\alpha )}
1025:is the assigned class label and
2954:Statistical charts and diagrams
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2857:
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68:, or that the population has a
2843:10.1080/15598608.2010.10411978
2184:Artificial Intelligence Review
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2046:
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2016:
2004:
1995:
1954:
1945:10.1080/00401706.1969.10490657
1924:
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928:
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901:
897:
555:for some nonnegative constant
532:
506:
484:
458:
1:
1918:
2315:Errata to the original paper
2228:Outliers in Statistical Data
1971:Introduction to Econometrics
1742:Set-membership uncertainties
1679:
1670:
605:indicates an "outlier", and
7:
2927:Encyclopedia of Mathematics
1861:
1810:that do not intersect the
246:. Others are model-based.
38:Michelson–Morley experiment
10:
2980:
2768:Dixon, W. J. (June 1960).
1868:Anomaly (natural sciences)
1388:{\displaystyle L\subset H}
692:
689:Modified Thompson Tau test
638:
579:proposed this test, where
301:
264:Grubbs's test for outliers
18:
2658:Karch, Julian D. (2023).
2604:10.1007/s10618-012-0300-z
2586:Schubert, E.; Zimek, A.;
2426:Exploratory Data Analysis
2253:Zimek, A.; Schubert, E.;
237:Definitions and detection
223:
105:probability distributions
70:heavy-tailed distribution
2940:described by NIST manual
2743:Wike, Edward L. (2006).
1852:hierarchical Bayes model
1758:is represented by a set
1726:Non-normal distributions
1631:trained on training set
1014:{\displaystyle 1-p(y|x)}
421:are the lower and upper
244:normal probability plots
21:Outlier (disambiguation)
2887:10.1049/ip-vis:19941330
2787:10.1214/aoms/1177705900
2389:Peirce, Charles Sanders
2366:Peirce, Charles Sanders
2022:Ripley, Brian D. 2004.
1898:Random sample consensus
1893:Influential observation
1748:set membership approach
1648:{\displaystyle \alpha }
878:-2 degrees of freedom,
545:{\displaystyle {\big }}
2624:Thompson .R. (1985). "
2423:Tukey, John W (1977).
1842:
1649:
1625:
1598:
1553:
1500:
1389:
1363:
1362:{\displaystyle p(h|t)}
1319:
1015:
952:
864:
821:
625:
599:
569:
546:
415:
388:
356:
343:
323:
165:
41:
2710:Psychological Methods
2563:10.1145/335191.335388
2513:10.1145/342009.335437
2482:10.1007/s007780050006
2230:(3 ed.), Wiley,
1820:
1659:Working with outliers
1650:
1635:with hyperparameters
1626:
1624:{\displaystyle g_{j}}
1599:
1554:
1470:
1390:
1364:
1320:
1016:
953:
865:
822:
626:
600:
598:{\displaystyle k=1.5}
570:
547:
416:
414:{\displaystyle Q_{3}}
389:
387:{\displaystyle Q_{1}}
354:2:558 by Chauvenet.)
352:A Manual of Astronomy
344:
324:
307:
259:Chauvenet's criterion
202:binomial distribution
163:
156:Occurrence and causes
138:than the rest of the
31:
2964:Statistical outliers
2311:Astronomical Journal
2165:; Leroy, A. (1996),
1908:Studentized residual
1888:Extreme value theory
1789:relaxed intersection
1639:
1608:
1566:
1402:
1373:
1336:
1044:
982:
886:
835:
707:
679:Local Outlier Factor
635:In anomaly detection
609:
583:
559:
432:
398:
371:
333:
313:
284:Mahalanobis distance
210:Poisson distribution
183:Poisson distribution
170:normally distributed
2822:Jaulin, L. (2010).
2809:Weisstein, Eric W.
2557:. pp. 93–104.
1736:Cauchy distribution
810:
683:k-nearest neighbors
624:{\displaystyle k=3}
365:interquartile range
82:normal distribution
2722:10.1037/met0000014
2676:10.1037/xge0001357
2429:. Addison-Wesley.
2029:2012-10-21 at the
1878:Anscombe's quartet
1846:Alternative models
1843:
1696:(or trimming) and
1645:
1621:
1594:
1549:
1385:
1359:
1315:
1313:
1263:
1170:
1094:
1011:
948:
947:
874:distribution with
860:
859:
817:
788:
621:
595:
565:
542:
411:
384:
367:. For example, if
339:
319:
304:Peirce's criterion
298:Peirce's criterion
178:standard deviation
166:
42:
2959:Robust statistics
2436:978-0-201-07616-5
2408:978-0-253-37201-7
2313:II 45 (1852) and
2271:10.1002/sam.11161
2237:978-0-471-93094-5
2110:10.1002/widm.1280
2075:978-3-540-26256-5
2024:Robust statistics
1985:978-0-02-374545-4
1903:Robust regression
1873:Novelty detection
1468:
1254:
1161:
1085:
815:
812:
771:
713:
675:anomaly detection
651:signal processing
641:Anomaly detection
568:{\displaystyle k}
342:{\displaystyle n}
322:{\displaystyle m}
66:measurement error
36:of data from the
2971:
2934:
2916:
2915:
2891:
2890:
2870:
2864:
2861:
2855:
2854:
2828:
2819:
2813:
2807:
2801:
2798:
2792:
2791:
2789:
2765:
2759:
2758:
2740:
2734:
2733:
2705:
2696:
2695:
2670:(6): 1735–1753.
2655:
2642:
2635:
2629:
2622:
2616:
2615:
2583:
2577:
2576:
2548:
2536:Breunig, M. M.;
2533:
2527:
2526:
2500:
2494:
2493:
2475:
2455:
2449:
2448:
2420:
2414:
2412:
2385:
2379:
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2362:
2356:
2355:
2344:10.2307/25138498
2328:Peirce, Benjamin
2324:
2318:
2300:
2294:
2289:
2283:
2282:
2250:
2241:
2240:
2223:
2217:
2216:
2199:
2179:
2173:
2172:
2159:
2153:
2150:
2144:
2143:
2141:
2140:
2134:
2128:. Archived from
2095:
2086:
2080:
2079:
2063:
2050:
2044:
2039:
2033:
2020:
2014:
2008:
2002:
1999:
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1958:
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1169:
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957:
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774:
772:
767:
764:
763:
762:
758:
742:
741:
740:
736:
721:
719:
714:
712:Rejection Region
711:
630:
628:
627:
622:
604:
602:
601:
596:
574:
572:
571:
566:
551:
549:
548:
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393:
391:
390:
385:
383:
382:
348:
346:
345:
340:
328:
326:
325:
320:
174:three sigma rule
150:central tendency
99:or flaws in the
97:systematic error
2979:
2978:
2974:
2973:
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2799:
2795:
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2762:
2755:
2741:
2737:
2706:
2699:
2656:
2645:
2636:
2632:
2623:
2619:
2588:Kriegel, H. -P.
2584:
2580:
2573:
2546:
2534:
2530:
2523:
2501:
2497:
2456:
2452:
2437:
2421:
2417:
2409:
2386:
2382:
2363:
2359:
2325:
2321:
2303:Benjamin Peirce
2301:
2297:
2290:
2286:
2251:
2244:
2238:
2224:
2220:
2197:10.1.1.109.1943
2180:
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2147:
2138:
2136:
2132:
2093:
2087:
2083:
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2031:Wayback Machine
2021:
2017:
2009:
2005:
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1959:
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1929:
1925:
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1864:
1848:
1809:
1782:
1775:
1764:
1744:
1728:
1713:Cook's distance
1682:
1673:
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1640:
1637:
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1632:
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204:with parameter
168:In the case of
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127:degrees Celsius
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5:
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2897:External links
2895:
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2881:(4): 217–222.
2865:
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2780:(2): 385–391.
2760:
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2716:(3): 409–427.
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2538:Kriegel, H.-P.
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2495:
2473:10.1.1.43.1842
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2435:
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2265:(5): 363–387.
2255:Kriegel, H.-P.
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673:, the task of
639:Main article:
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359:Tukey's fences
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302:Main article:
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250:are a hybrid.
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116:sample minimum
112:sample maximum
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2904:Renze, John.
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2572:1-58113-217-4
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2560:
2556:
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2222:
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2203:
2198:
2193:
2190:(2): 85–126,
2189:
2185:
2178:
2170:
2169:
2164:
2158:
2149:
2135:on 2021-11-14
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1934:
1933:Technometrics
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1866:
1865:
1859:
1857:
1856:mixture model
1853:
1841:= 5 (yellow).
1840:
1836:
1832:
1828:
1824:
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1802:
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1720:data analysis
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663:manufacturing
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194:significantly
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87:
86:mixture model
83:
79:
75:
71:
67:
62:
60:
55:
51:
47:
39:
35:
30:
26:
22:
2925:
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2874:
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2859:
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2796:
2777:
2773:
2763:
2744:
2738:
2713:
2709:
2667:
2663:
2633:
2620:
2595:
2591:
2581:
2550:
2542:
2531:
2504:
2498:
2466:(3–4): 237.
2463:
2459:
2453:
2425:
2418:
2393:
2383:
2369:
2360:
2335:
2331:
2322:
2310:
2298:
2287:
2262:
2258:
2227:
2221:
2187:
2183:
2177:
2167:
2163:Rousseeuw, P
2157:
2148:
2137:. Retrieved
2130:the original
2104:(6): e1280.
2101:
2097:
2084:
2059:
2048:
2037:
2018:
2006:
1997:
1989:
1970:
1956:
1948:
1936:
1932:
1926:
1849:
1838:
1837:= 4 (blue),
1834:
1833:=3 (green),
1830:
1826:
1822:
1811:
1804:
1800:
1796:
1792:
1784:
1777:
1770:
1766:
1759:
1755:
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1729:
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1706:
1691:
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1662:
1561:
1327:
976:
970:
968:
963:
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830:
698:
674:
659:econometrics
644:
554:
362:
351:
308:
270:
252:
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217:
213:
205:
197:
189:
187:
167:
144:
120:
109:
90:
63:
49:
43:
25:
2938:Grubbs test
2837:: 155–167.
2598:: 190–237.
2338:: 348–351.
2011:Grubbs 1969
1939:(1): 1–21.
1913:Winsorizing
1698:Winsorising
671:data mining
231:King effect
93:sample mean
2948:Categories
2522:1581132174
2376:PDF Eprint
2372:: 200–224.
2139:2019-12-11
1966:"Outliers"
1919:References
1829:=2 (red),
1821:Figure 5.
1709:regression
1694:truncation
1676:outliers.
693:See also:
667:networking
647:statistics
577:John Tukey
172:data, the
146:Estimators
136:population
54:data point
46:statistics
32:Figure 1.
2932:EMS Press
2922:"Outlier"
2911:MathWorld
2906:"Outlier"
2692:258376426
2468:CiteSeerX
2192:CiteSeerX
2118:1942-4787
1732:fat tails
1680:Exclusion
1671:Retention
1643:α
1589:α
1541:α
1472:∑
1446:−
1434:⟩
1422:⟨
1380:⊂
1256:∑
1252:−
1192:−
1163:∑
1102:−
1087:∑
1073:⟩
1061:⟨
989:−
908:−
891:δ
846:α
794:α
779:−
753:−
730:α
520:−
472:−
453:−
423:quartiles
248:Box plots
212:with λ =
2851:16500768
2730:24773354
2684:37104797
2612:19036098
2490:11707259
2352:25138498
2126:53305944
2054:Ruan, Da
2027:Archived
1964:(1992).
1862:See also
1702:censored
288:leverage
269:Dixon's
242:such as
190:a priori
78:skewness
59:data set
34:Box plot
2445:3058187
2399:140–160
2374:. NOAA
2279:6724536
2214:3330313
1854:, or a
655:finance
123:average
50:outlier
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2555:SIGMOD
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1982:
1704:data.
1562:where
1021:where
831:where
224:Causes
140:sample
131:median
101:theory
74:robust
2847:S2CID
2827:(PDF)
2688:S2CID
2608:S2CID
2547:(PDF)
2486:S2CID
2348:JSTOR
2275:S2CID
2210:S2CID
2133:(PDF)
2122:S2CID
2094:(PDF)
958:. If
142:set.
52:is a
48:, an
2749:ISBN
2726:PMID
2680:PMID
2567:ISBN
2517:ISBN
2441:OCLC
2431:ISBN
2403:ISBN
2232:ISBN
2114:ISSN
2070:ISBN
1980:ISBN
669:and
394:and
286:and
278:ASTM
273:test
2883:doi
2879:141
2839:doi
2782:doi
2718:doi
2672:doi
2668:152
2600:doi
2559:doi
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2478:doi
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2106:doi
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1941:doi
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