Figure 1. Box plot of data from the Michelson-Morley Experiment displaying outliers in the middle column.

In statistics, an outlier^[1] is an observation that is numerically distant from the rest of the data. Grubbs^[2] defined an outlier as:

An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs.

Outliers can occur by chance in any distribution, but they are often indicative either of measurement error or that the population has a heavy-tailed distribution. In the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high kurtosis and that one should be very cautious in using tools or intuitions that assume a normal distribution. 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 mixture model.

In most larger samplings of data, some data points will be further away from the sample mean than what is deemed reasonable. This can be due to incidental systematic error or flaws in the theory that generated an assumed family of probability distributions, 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).

Outliers, being the most extreme observations, may include the sample maximum or sample minimum, 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.

Naive interpretation of statistics derived from data sets that include outliers may be misleading. For example, if one is calculating the average temperature of 10 objects in a room, and most are between 20 and 25 degrees Celsius, but an oven is at 175 °C, the median of the data may be 23 °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 than the mean; however, naively interpreting the mean as "a typical sample", equivalent to the median, is incorrect. As illustrated in this case, outliers may be indicative of data points that belong to a different population than the rest of the sample set.

Estimators capable of coping with outliers are said to be robust: the median is a robust statistic, while the mean is not.

Occurrence and causes[link]

In the case of normally distributed data, roughly 1 in 22 observations will differ by twice the standard deviation or more from the mean, and 1 in 370 will deviate by three times the standard deviation; see three sigma rule^[3] for details. 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 the expected number and hence within 1 standard deviation of the expected number – see Poisson distribution, and not indicative of 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.

In general, if the nature of the population distribution is known a priori, it is possible to test if the number of outliers deviate significantly from what can be expected: for a given cutoff (so samples fall beyond the cutoff with probability p) of a given distribution, the number of outliers will follow a binomial distribution with parameter p, which can generally be well-approximated by the Poisson distribution with λ = pn. Thus if one takes a normal distribution with cutoff 3 standard deviations from the mean, p is approximately .3%, and thus for 1,000 trials one can approximate the number of samples whose deviation exceeds 3 sigmas by a Poisson distribution with λ = 3.

Causes[link]

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 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 (King effect).

Caution[link]

Unless it can be ascertained that the deviation is not significant, it is ill-advised to ignore the presence of outliers. Outliers that cannot be readily explained demand special attention – see kurtosis risk and black swan theory.

Identifying outliers[link]

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.

Outlier detection^[4] has been used for centuries to detect and, where appropriate, remove anomalous observations from data. Outlier detection can identify system faults and fraud before they escalate with potentially catastrophic consequences. The original outlier detection methods were arbitrary but now, principled and systematic techniques are used, drawn from the full gamut of computer science and statistics.

There are three fundamental approaches to the problem of outlier detection:

• Type 1 - Determine the outliers with no prior knowledge of the data. This is essentially a learning approach analogous to unsupervised clustering. The approach processes the data as a static distribution, pinpoints the most remote points, and flags them as potential outliers.
• Type 2 - Model both normality and abnormality. This approach is analogous to supervised classification and requires pre-labeled data, tagged as normal or abnormal.
• Type 3 - Model only normality (or in a few cases model abnormality). This is analogous to a semi-supervised recognition or detection task. It may be considered semi-supervised as the normal class is taught but the algorithm learns to recognize abnormality.

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:

• Chauvenet's criterion
• Grubbs' test for outliers
• Peirce's criterion^[5][6][7][8]

It is proposed to determine in a series of Failed to parse (Missing texvc executable; please see math/README to configure.): m observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as Failed to parse (Missing texvc executable; please see math/README to configure.): n 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 probability of making so many, and no more, abnormal observations. (Quoted in the editorial note on page 516 to Peirce (1982 edition) from A Manual of Astronomy 2:558 by Chauvenet.)

• Dixon's Q test
• ASTM E178 Standard Practice for Dealing With Outlying Observations

Other methods flag observations based on measures such as the interquartile range. For example, if Failed to parse (Missing texvc executable; please see math/README to configure.): Q_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): Q_3
are the lower and upper quartiles respectively, then one could define an outlier to be any observation outside the range:

Failed to parse (Missing texvc executable; please see math/README to configure.): \big[ Q_1 - k (Q_3 - Q_1 ) , Q_3 + k (Q_3 - Q_1 ) \big]

for some constant Failed to parse (Missing texvc executable; please see math/README to configure.): k .

Other approaches are distance-based^[9][10] and frequently use the distance to the k-nearest neighbors to label observations as outliers or non-outliers.

Working with outliers[link]

The choice of how to deal with an outlier should depend on the cause.

Retention[link]

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. The application should use a classification algorithm that is robust to outliers to model data with naturally occurring outlier points.

Exclusion[link]

Deletion of outlier data is a controversial practice frowned on by many scientists and science instructors; while mathematical criteria provide an objective and quantitative method for data 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. An outlier resulting from an instrument reading error may be excluded but it is desirable that the reading is at least verified.

In regression problems, an alternative approach may be to only exclude points which exhibit a large degree of influence on the parameters, using a measure such as Cook's distance.^[11]

If a data point (or points) is excluded from the data analysis, this should be clearly stated on any subsequent report.

Non-normal distributions[link]

The possibility should be considered that the underlying distribution of the data is not approximately normal, having "fat tails". For instance, when sampling from a Cauchy distribution,^[12] 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.

Alternative models[link]

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 hierarchical Bayes model or a mixture model.^[13][14]

See also[link]

• Anomaly detection
• Anomaly time series
• Robust statistics
• Robust regression
• Box plot
• King effect
• Studentized residual
• Winsorising
• Data transformation (statistics)

References[link]

• ISO 16269-4, Standard Interpretation of Data, Part 4, Determination of Outliers
• Strutz, Tilo (2010). Data Fitting and Uncertainty - A practical introduction to weighted least squares and beyond. Vieweg+Teubner. ISBN 978-3-8348-1022-9. 

External links[link]

• Renze, John, "Outlier" from MathWorld.
• Balakrishnan, N.; Childs, A. (2001), "Outlier", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=O/o110080 
• Grubbs test described by NIST manual
• how to detect univariate outliers, how to detect multivariate outliers and how to deal with outliers

Malcolm Gladwell
Gladwell at PopTech!, October 2008
Born	Malcolm T. Gladwell (1963-09-03) September 3, 1963 (age 48) Fareham, Hampshire, United Kingdom
Occupation	Non-fiction writer, journalist
Nationality	Canadian
Period	1987–present
Notable work(s)	The Tipping Point (2000) Blink (2005) Outliers (2008) What the Dog Saw: And Other Adventures (2009)

Malcolm T. Gladwell, CM (born September 3, 1963) is a Canadian journalist, bestselling author, and speaker.^[1] He is currently based in New York City and has been a staff writer for The New Yorker since 1996. He has written four books, The Tipping Point: How Little Things Make a Big Difference (2000), Blink: The Power of Thinking Without Thinking (2005), Outliers: The Story of Success (2008), and What the Dog Saw: And Other Adventures (2009), a collection of his journalism. All four books were New York Times Bestsellers.

Gladwell's books and articles often deal with the unexpected implications of research in the social sciences and make frequent and extended use of academic work, particularly in the areas of sociology, psychology, and social psychology. Gladwell was appointed to the Order of Canada on June 30, 2011.^[2]

Early life[link]

Gladwell was born in Fareham, Hampshire, England to Joyce, Jamaican-born psychotherapist, and Graham Gladwell, a British mathematics professor.^[3][4] Gladwell has said that his mother is his role model as a writer.^[5] When he was six his family moved to Elmira, Ontario, Canada.^[3]

Gladwell’s father noted that Malcolm was an unusually single-minded and ambitious boy.^[6] When Malcolm was 11, his father, who was a professor^[7] of mathematics and engineering at the University of Waterloo, allowed him to wander around the offices at his university, which stoked the boy's interest in reading and libraries.^[8] During his high school years, Gladwell was an outstanding middle-distance runner and won the 1500 meter title at the 1978 Ontario High School 14 year old championships in Kingston, Ontario.^[9] In the spring of 1982, Gladwell interned with the National Journalism Center in Washington, D.C.^[10] He graduated with a degree in history from the University of Toronto's Trinity College in 1984.^[11]

Career[link]

Gladwell’s grades weren’t good enough for graduate school (as Gladwell puts it, “college was not an... intellectually fruitful time for me”), so he decided to go into advertising.^[8][12] After being rejected by every advertising agency he applied to, he accepted a journalism position at The American Spectator and moved to Indiana.^[13] He subsequently wrote for Insight on the News, a conservative magazine owned by the Rev. Sun Myung Moon's Unification Church.^[14] In 1987, Gladwell began covering business and science for The Washington Post, where he worked until 1996.^[15] In a personal elucidation of the 10,000 hour rule he popularized in Outliers, Gladwell notes, "I was a basket case at the beginning, and I felt like an expert at the end. It took 10 years — exactly that long."^[8]

When he started at The New Yorker in 1996 he wanted to "mine current academic research for insights, theories, direction, or inspiration."^[6] His first assignment was to write a piece about fashion. Instead of writing about high-class fashion, Gladwell opted to write a piece about a man who manufactured T-shirts, saying “it was much more interesting to write a piece about someone who made a T-shirt for $8 than it was to write about a dress that costs $100,000. I mean, you or I could make a dress for $100,000, but to make a T-shirt for $8 – that’s much tougher.”^[6] Gladwell gained popularity with two New Yorker articles, both written in 1996: "The Tipping Point"^[16] and "The Coolhunt"^[17][18] These two pieces would become the basis for Gladwell's first book, The Tipping Point, for which he received a $1 million advance.^[12][19] He continues to write for The New Yorker. He also serves as a contributing editor for Grantland, a sports journalism website founded by ESPN's Bill Simmons.

Works[link]

Gladwell has written four books. When asked for the process behind his writing, he said "I have two parallel things I'm interested in. One is, I'm interested in collecting interesting stories, and the other is I'm interested in collecting interesting research. What I'm looking for is cases where they overlap."^[20] The initial inspiration for his first book, The Tipping Point, came from the sudden drop of crime in New York City.^[21] He wanted the book to have a broader appeal than just crime, however, and sought to explain similar phenomena through the lens of epidemiology. While Gladwell was a reporter for The Washington Post, he covered the AIDS epidemic. He began to take note of "how strange epidemics were," saying that epidemiologists have a "strikingly different way of looking at the world."^[22] The word "tipping point" comes from the moment in an epidemic when the virus reaches critical mass and begins to spread at a much higher rate.^[22]

After the success of The Tipping Point, Gladwell wrote Blink in 2005. The book explains how the human subconscious interprets events or cues and how past experiences can lead people to make informed decisions very rapidly, using examples like the Getty kouros and psychologist John Gottman's research on the likelihood of divorce in married couples. Gladwell’s hair was the inspiration for Blink.^[23] He stated that he started to get speeding tickets all the time, an oddity considering that he had never got one before, and that he started getting pulled out of airport security lines for special attention.^[24] In a particular incident, he was accosted by three police officers while walking in downtown Manhattan, because his curly hair matched the profile of a rapist, despite the fact that the suspect looked nothing like him otherwise.^[25]

Gladwell’s third book, Outliers, published in 2008, examines how a person's environment, in conjunction with personal drive and motivation, affects his or her possibility and opportunity for success. Gladwell’s original question revolved around lawyers: "We take it for granted that there’s this guy in New York who’s the corporate lawyer, right? I just was curious: Why is it all the same guy?"^{[clarification needed][8]} In another example present in the book, Gladwell noticed that people ascribe Bill Gates’s success to being "really smart" or "really ambitious." He noted that he knew a lot of people who are really smart and really ambitious, but not worth 60 billion dollars. "It struck me that our understanding of success was really crude – and there was an opportunity to dig down and come up with a better set of explanations."^[26]

Gladwell's fourth book, What the Dog Saw: And Other Adventures, was published on October 20, 2009.^[27] What the Dog Saw bundles together his favorite articles from The New Yorker since he joined the magazine as a staff writer in 1996.^[28] The stories share a common theme, namely that Gladwell tries to show us the world through the eyes of others, even if that other happens to be a dog.^[29][30]

Gladwell's books The Tipping Point (2000) and Blink (2005), were international bestsellers. The Tipping Point sold over two million copies in the United States. Blink sold equally well.^[12][31]

Reception[link]

The Tipping Point was named as one of the best books of the decade by Amazon.com customers, The Onion A.V. Club, The Guardian, and The Times.^{[32][33][34][35]} It was also Barnes and Nobles’s 5th bestselling nonfiction book of the decade.^[36] Blink was named to Fast Company’s list of the best business books of 2005.^[37] It was also #5 on Amazon users’ favorite books of 2005, named to Christian Science Monitor’s best nonfiction books of 2005, and in the top 50 of Amazon users’ favorite books of the decade.^[32][38][39] Outliers was a #1 New York Times Bestseller for 11 straight weeks, and was Time’s #10 nonfiction book of 2008, as well as named to The San Francisco Chronicle’s list of the 50 best nonfiction books of 2008.^[40][41][42]

Critical appraisal of Gladwell's work has been mixed. Most praise his gift for compelling writing and clarity of expression while many disagree with his conclusions or question the validity of his methods.

Fortune described The Tipping Point as “a fascinating book that makes you see the world in a different way.”^[43][44] The Daily Telegraph called it “a wonderfully offbeat study of that little-understood phenomenon, the social epidemic.”^[45] Reviewing Blink, the Baltimore Sun dubbed Gladwell “the most original American [sic] journalist since the young Tom Wolfe.”^[46] Farhad Manjoo at Salon described the book as “a real pleasure. As in the best of Gladwell's work, Blink brims with surprising insights about our world and ourselves.”^[47] The Economist called Outliers “a compelling read with an important message.”^[48] David Leonhardt wrote in The New York Times Book Review: “In the vast world of nonfiction writing, Malcolm Gladwell is as close to a singular talent as exists today” and that Outliers “leaves you mulling over its inventive theories for days afterward.”^[49] Ian Sample wrote in the Guardian: “Brought together, the pieces form a dazzling record of Gladwell's art. There is depth to his research and clarity in his arguments, but it is the breadth of subjects he applies himself to that is truly impressive.”^[50][51]

Criticism of Gladwell tends to focus on the fact that he is a journalist and not a scientist, and as a result his work is prone to oversimplification and logical fallacies. Critics charge that he sometimes stretches his colorful stories to make them apply to business issues.^[52] The New Republic called the final chapter of Outliers, "impervious to all forms of critical thinking".^[53] Gladwell has also received criticism for his emphasis on anecdotal evidence over research to support his conclusions.^[54] Maureen Tkacik and Steven Pinker have challenged the integrity of Gladwell's approach.^[55][56] Even while praising Gladwell's attractive writing style and content, Pinker sums up his take on Gladwell as, "a minor genius who unwittingly demonstrates the hazards of statistical reasoning," while accusing Gladwell of "cherry-picked anecdotes, post-hoc sophistry and false dichotomies" in his book Outliers. Referencing a Gladwell reporting mistake, Pinker criticizes his lack of expertise: "I will call this the Igon Value [sic] Problem: when a writer’s education on a topic consists in interviewing an expert, he is apt to offer generalizations that are banal, obtuse or flat wrong."^[55] A writer in The Independent accused Gladwell of posing "obvious" insights.^[57] The Register has accused Gladwell of making arguments by weak analogy and commented that Gladwell has an "aversion for fact", adding that, "Gladwell has made a career out of handing simple, vacuous truths to people and dressing them up with flowery language and an impressionistic take on the scientific method."^[58] Gladwell's approach has been satirized by the online site "The Malcolm Gladwell Book Generator".^[59]

In 2012, CBS's 60 Minutes attributed the recent trend of American parents "redshirting" their five-year-olds (postponing entrance) to give them an advantage in kindergarten to a section in Gladwell's Outliers.^[60]

On May 31, 2012, Malcolm Gladwell was the subject of an investigative report documenting his professional origins and financial relationships with various industry groups. These profitable, long-term business relationships appear to have connection with journalistic stances presented by Gladwell as original and unbiased. Authored by Yasha Levine of The eXiled, and titled "Malcolm Gladwell Unmasked: A Look Into the Life & Work of America’s Most Successful Propagandist," the exposé alleges a seedier, blatantly corrupt dimension to existing criticisms of Gladwell's work. ^[61]

Awards and honors[link]

• 2005 Time named Gladwell one of its 100 most influential people.^[62]
• 2007 American Sociological Association's first Award for Excellence in the Reporting of Social Issues.^[63]
• 2007 honorary degree from University of Waterloo.^[64][65]
• 2011 honorary degree from University of Toronto

Bibliography[link]

• The Tipping Point (2000)
• Blink (2005)
• Outliers (2008)
• What the Dog Saw: And Other Adventures (2009)

See also[link]

	Biography portal
	United States portal
	Books portal

• Nassim Nicholas Taleb
• Stephen J. Dubner
• Chris Anderson

References[link]

External links[link]

• Gladwell's personal website
• Biography and list of articles at The New Yorker
• Malcolm Gladwell at TED Conferences
• Appearances on C-SPAN
• Malcolm Gladwell on Charlie Rose
• Malcolm Gladwell at the Internet Movie Database
• Works by or about Malcolm Gladwell in libraries (WorldCat catalog)
• Malcolm Gladwell collected news and commentary at The Guardian
• Book Review in the New York Times
• Book Review in The Nation
• Malcolm Gladwell Interview New Statesman May 2010