"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)
"Logging skewed variables also helps to reveal the patterns in the data. […] the rescaling of the variables by taking logarithms reduces the nonlinearity in the relationship and removes much of the clutter resulting from the skewed distributions on both variables; in short, the transformation helps clarify the relationship between the two variables. It also […] leads to a theoretically meaningful regression coefficient." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)
"The logarithmic transformation serves several purposes: (1) The resulting regression coefficients sometimes have a more useful theoretical interpretation compared to a regression based on unlogged variables. (2) Badly skewed distributions - in which many of the observations are clustered together combined with a few outlying values on the scale of measurement - are transformed by taking the logarithm of the measurements so that the clustered values are spread out and the large values pulled in more toward the middle of the distribution. (3) Some of the assumptions underlying the regression model and the associated significance tests are better met when the logarithm of the measured variables is taken." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)
"The logarithm is an extremely powerful and useful tool for graphical data presentation. One reason is that logarithms turn ratios into differences, and for many sets of data, it is natural to think in terms of ratios. […] Another reason for the power of logarithms is resolution. Data that are amounts or counts are often very skewed to the right; on graphs of such data, there are a few large values that take up most of the scale and the majority of the points are squashed into a small region of the scale with no resolution." (William S. Cleveland, "Graphical Methods for Data Presentation: Full Scale Breaks, Dot Charts, and Multibased Logging", The American Statistician Vol. 38 (4) 1984)
"It is common for positive data to be skewed to the right: some values bunch together at the low end of the scale and others trail off to the high end with increasing gaps between the values as they get higher. Such data can cause severe resolution problems on graphs, and the common remedy is to take logarithms. Indeed, it is the frequent success of this remedy that partly accounts for the large use of logarithms in graphical data display." (William S Cleveland, "The Elements of Graphing Data", 1985)
"If a distribution were perfectly symmetrical, all symmetry-plot points would be on the diagonal line. Off-line points indicate asymmetry. Points fall above the line when distance above the median is greater than corresponding distance below the median. A consistent run of above-the-line points indicates positive skew; a run of below-the-line points indicates negative skew." (Lawrence C Hamilton, "Regression with Graphics: A second course in applied statistics", 1991)
"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)
"Data that are skewed toward large values occur commonly. Any set of positive measurements is a candidate. Nature just works like that. In fact, if data consisting of positive numbers range over several powers of ten, it is almost a guarantee that they will be skewed. Skewness creates many problems. There are visualization problems. A large fraction of the data are squashed into small regions of graphs, and visual assessment of the data degrades. There are characterization problems. Skewed distributions tend to be more complicated than symmetric ones; for example, there is no unique notion of location and the median and mean measure different aspects of the distribution. There are problems in carrying out probabilistic methods. The distribution of skewed data is not well approximated by the normal, so the many probabilistic methods based on an assumption of a normal distribution cannot be applied." (William S Cleveland, "Visualizing Data", 1993)
"The logarithm is one of many transformations that we can apply to univariate measurements. The square root is another. Transformation is a critical tool for visualization or for any other mode of data analysis because it can substantially simplify the structure of a set of data. For example, transformation can remove skewness toward large values, and it can remove monotone increasing spread. And often, it is the logarithm that achieves this removal." (William S Cleveland, "Visualizing Data", 1993)
"When the distributions of two or more groups of univariate data are skewed, it is common to have the spread increase monotonically with location. This behavior is monotone spread. Strictly speaking, monotone spread includes the case where the spread decreases monotonically with location, but such a decrease is much less common for raw data. Monotone spread, as with skewness, adds to the difficulty of data analysis. For example, it means that we cannot fit just location estimates to produce homogeneous residuals; we must fit spread estimates as well. Furthermore, the distributions cannot be compared by a number of standard methods of probabilistic inference that are based on an assumption of equal spreads; the standard t-test is one example. Fortunately, remedies for skewness can cure monotone spread as well." (William S Cleveland, "Visualizing Data", 1993)
"The standard deviation (often SD) is a measure of variability. When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. For data with a normal distribution, about 95% of individu als will have values within 2 standard deviations of the mean, the other 5% being equally scattered above and below these limits. Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. About 95% of observa tions of any distribution usually fall within the 2 standard deviation limits, though those outside may all be at one end. We may choose a different summary statistic, how ever, when data have a skewed distribution." (Douglas G Altman & J Martin Bland, "Statistics Notes: Standard Deviations And Standard Errors", British Medical Journal Vol. 331 (7521) 2005)
"Use a logarithmic scale when it is important to understand percent change or multiplicative factors. […] Showing data on a logarithmic scale can cure skewness toward large values." (Naomi B Robbins, "Creating More effective Graphs", 2005)
"Distributional shape is an important attribute of data, regardless of whether scores are analyzed descriptively or inferentially. Because the degree of skewness can be summarized by means of a single number, and because computers have no difficulty providing such measures (or estimates) of skewness, those who prepare research reports should include a numerical index of skewness every time they provide measures of central tendency and variability." (Schuyler W Huck, "Statistical Misconceptions", 2008)
"Given the important role that correlation plays in structural equation modeling, we need to understand the factors that affect establishing relationships among multivariable data points. The key factors are the level of measurement, restriction of range in data values (variability, skewness, kurtosis), missing data, nonlinearity, outliers, correction for attenuation, and issues related to sampling variation, confidence intervals, effect size, significance, sample size, and power." (Randall E Schumacker & Richard G Lomax, "A Beginner’s Guide to Structural Equation Modeling" 3rd Ed., 2010)
"[The normality] assumption is the least important one for the reliability of the statistical procedures under discussion. Violations of the normality assumption can be divided into two general forms: Distributions that have heavier tails than the normal and distributions that are skewed rather than symmetric. If data is skewed, the formulas we are discussing are still valid as long as the sample size is sufficiently large. Although the guidance about 'how skewed' and 'how large a sample' can be quite vague, since the greater the skew, the larger the required sample size. For the data commonly used in time series and for the sample sizes (which are generally quite large) used, skew is not a problem. On the other hand, heavy tails can be very problematic." (DeWayne R Derryberry, "Basic Data Analysis for Time Series with R" 1st Ed, 2014)
"In statistical theory, location and variability are referred to as the first and second moments of a distribution. The third and fourth moments are called skewness and kurtosis. Skewness refers to whether the data is skewed to larger or smaller values and kurtosis indicates the propensity of the data to have extreme values. Generally, metrics are not used to measure skewness and kurtosis; instead, these are discovered through visual displays [...]" (Peter C Bruce & Andrew G Bruce, "Statistics for Data Scientists: 50 Essential Concepts", 2016)
"A histogram represents the frequency distribution of the data. Histograms are similar to bar charts but group numbers into ranges. Also, a histogram lets you show the frequency distribution of continuous data. This helps in analyzing the distribution (for example, normal or Gaussian), any outliers present in the data, and skewness." (Umesh R Hodeghatta & Umesha Nayak, "Business Analytics Using R: A Practical Approach", 2017)
"New information is constantly flowing in, and your brain is constantly integrating it into this statistical distribution that creates your next perception (so in this sense 'reality' is just the product of your brain’s ever-evolving database of consequence). As such, your perception is subject to a statistical phenomenon known in probability theory as kurtosis. Kurtosis in essence means that things tend to become increasingly steep in their distribution [...] that is, skewed in one direction. This applies to ways of seeing everything from current events to ourselves as we lean 'skewedly' toward one interpretation, positive or negative. Things that are highly kurtotic, or skewed, are hard to shift away from. This is another way of saying that seeing differently isn’t just conceptually difficult - it’s statistically difficult." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)
"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)
"With skewed data, quantiles will reflect the skew, while adding standard deviations assumes symmetry in the distribution and can be misleading." (Robert Grant, "Data Visualization: Charts, Maps and Interactive Graphics", 2019)
"Adjusting scale is an important practice in data visualization. While the log transform is versatile, it doesn’t handle all situations where skew or curvature occurs. For example, at times the values are all roughly the same order of magnitude and the log transformation has little impact. Another transformation to consider is the square root transformation, which is often useful for count data." (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)
