"Some distributions [...] are symmetrical about their central value. Other distributions have marked asymmetry and are said to be skew. Skew distributions are divided into two types. If the 'tail' of the distribution reaches out into the larger values of the variate, the distribution is said to show positive skewness; if the tail extends towards the smaller values of the variate, the distribution is called negatively skew." (Michael J Moroney,Facts from Figures", 1951)
"Logging size transforms the original skewed distribution into a more symmetrical one by pulling in the long right tail of the distribution toward the mean. The short left tail is, in addition, stretched. The shift toward symmetrical distribution produced by the log transform is not, of course, merely for convenience. Symmetrical distributions, especially those that resemble the normal distribution, fulfill statistical assumptions that form the basis of statistical significance testing in the regression model." (Edward R Tufte,Data Analysis for Politics and Policy", 1974)
"Equal variability is not always achieved in plots. For instance, if the theoretical distribution for a probability plot has a density that drops off gradually to zero in the tails" (as the normal density does), then the variability of the data in the tails of the probability plot is greater than in the center. Another example is provided by the histogram. Since the height of any one bar has a binomial distribution, the standard deviation of the height is approximately proportional to the square root of the expected height; hence, the variability of the longer bars is greater." (John M Chambers et al,Graphical Methods for Data Analysis", 1983)
"If the sample is not representative of the population because the sample is small or biased, not selected at random, or its constituents are not independent of one another, then the bootstrap will fail. […] For a given size sample, bootstrap estimates of percentiles in the tails will always be less accurate than estimates of more centrally located percentiles. Similarly, bootstrap interval estimates for the variance of a distribution will always be less accurate than estimates of central location such as the mean or median because the variance depends strongly upon extreme values in the population." (Phillip I Good & James W Hardin,Common Errors in Statistics" (and How to Avoid Them)", 2003)
"Bell curves don't differ that much in their bells. They differ in their tails. The tails describe how frequently rare events occur. They describe whether rare events really are so rare. This leads to the saying that the devil is in the tails." (Bart Kosko,Noise", 2006)
"Readability in visualization helps people interpret data and make conclusions about what the data has to say. Embed charts in reports or surround them with text, and you can explain results in detail. However, take a visualization out of a report or disconnect it from text that provides context" (as is common when people share graphics online), and the data might lose its meaning; or worse, others might misinterpret what you tried to show." (Nathan Yau,Data Points: Visualization That Means Something", 2013)
"A very different - and very incorrect - argument is that successes must be balanced by failures (and failures by successes) so that things average out. Every coin flip that lands heads makes tails more likely. Every red at roulette makes black more likely. […] These beliefs are all incorrect. Good luck will certainly not continue indefinitely, but do not assume that good luck makes bad luck more likely, or vice versa." (Gary Smith,Standard Deviations", 2014)
"The more complex the system, the more variable (risky) the outcomes. The profound implications of this essential feature of reality still elude us in all the practical disciplines. Sometimes variance averages out, but more often fat-tail events beget more fat-tail events because of interdependencies. If there are multiple projects running, outlier (fat-tail) events may also be positively correlated - one IT project falling behind will stretch resources and increase the likelihood that others will be compromised." (Paul Gibbons,The Science of Successful Organizational Change", 2015)
"Many statistical procedures perform more effectively on data that are normally distributed, or at least are symmetric and not excessively kurtotic" (fat-tailed), and where the mean and variance are approximately constant. Observed time series frequently require some form of transformation before they exhibit these distributional properties, for in their 'raw' form they are often asymmetric." (Terence C Mills,Applied Time Series Analysis: A practical guide to modeling and forecasting", 2019)
"Mean-averages can be highly misleading when the raw data do not form a symmetric pattern around a central value but instead are skewed towards one side [...], typically with a large group of standard cases but with a tail of a few either very high" (for example, income) or low" (for example, legs) values." (David Spiegelhalter,The Art of Statistics: Learning from Data", 2019)
"[…] it is not merely that events in the tails of the distributions matter, happen, play a large role, etc. The point is that these events play the major role and their probabilities are not" (easily) computable, not reliable for any effective use. The implication is that Black Swans do not necessarily come from fat tails; the problem can result from an incomplete assessment of tail events." (Nassim N Taleb,Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)
"[…] whenever people make decisions after being supplied with the standard deviation number, they act as if it were the expected mean deviation." (Nassim N Taleb,Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)
"Behavioral finance so far makes conclusions from statics not dynamics, hence misses the picture. It applies trade-offs out of context and develops the consensus that people irrationally overestimate tail risk" (hence need to be 'nudged' into taking more of these exposures). But the catastrophic event is an absorbing barrier. No risky exposure can be analyzed in isolation: risks accumulate. If we ride a motorcycle, smoke, fly our own propeller plane, and join the mafia, these risks add up to a near-certain premature death. Tail risks are not a renewable resource." (Nassim N Taleb,Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)
"But note that any heavy tailed process, even a power law, can be described in sample" (that is finite number of observations necessarily discretized) by a simple Gaussian process with changing variance, a regime switching process, or a combination of Gaussian plus a series of variable jumps" (though not one where jumps are of equal size […])." (Nassim N Taleb,Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)
"Once we know something is fat-tailed, we can use heuristics to see how an exposure there reacts to random events: how much is a given unit harmed by them. It is vastly more effective to focus on being insulated from the harm of random events than try to figure them out in the required details" (as we saw the inferential errors under thick tails are huge). So it is more solid, much wiser, more ethical, and more effective to focus on detection heuristics and policies rather than fabricate statistical properties." (Nassim N Taleb,Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)
"No one sees further into a generalization than his own knowledge of detail extends." (William James)
"Remember that a p-value merely indicates the probability of a particular set of data being generated by the null model–it has little to say about the size of a deviation from that model" (especially in the tails of the distribution, where large changes in effect size cause only small changes in p-values)." (Clay Helberg)
