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I'm curious about the nomenclature: why is left-skewed called negatively skewed and right-skewed called positively skewed?

Graphs depicting positive and negative skew

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    $\begingroup$ Let's underline that the terms left and right depend on a tacit convention that the magnitude axis of a graph showing a distribution s horizontal with negative values to the left. This may seem too obvious to state, except to those who do things differently. $\endgroup$ – Nick Cox Apr 11 '16 at 15:32
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My short answer is that it is by design. The skewness measures are usually constructed so that the positive skewness indicates right-skewed distributions.

Today the most common measure of skewness, that is also usually taught in schools, is based on the third central moment equation as follows:

$$\mu_3=E[(X-\mu)^3]$$

Look at the expression above. When there's more weight (of the distribution function) to the right of the mean then $(x-\mu)^3$ will contribute more positive values. The right of the mean is positive, because $x>\mu$ and the left is negative because $x<\mu$. So, mechanically it would seem to answer exactly your question.

However, as @Nick Cox brought up, there is more than one measure of skewness, such as Pearson's first coefficient of skewness, which is based on the difference $mean-mode$. Potentially, different measures of skewness could lead to different relations between positive skewness and the tendency to have heavier tails on the right.

Hence, it is interesting to look at why these measures of skewness were introduced in the first place, and why do they have their particular formulations.

In this context it is useful to look at the exposition of skewness by Yule in An Introduction to the Theory of Statistics (1912). In the following excerpt he describes the desired properties of a reasonable skewness measure. Basically, he requires that the positive skewness should correspond to right skewed distributions, like in your picture:

enter image description here

enter image description here

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    $\begingroup$ Correct, but incomplete in so far as there are several other ways to measure skewness. But all I that know of follow the same convention that right-skewed and left-skewed typically yield positive and negative results respectively, as for example (mean $-$ median)/SD. The only certain thing, however, is that symmetrical distributions have zero skewness. It is possible to have asymmetrical distributions for which different skewness measures don't even agree in sign. $\endgroup$ – Nick Cox Apr 11 '16 at 15:30
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    $\begingroup$ I believe you, but the question remains general and benefits from a general answer. In a century or so, considerable confusion has already been caused by conflating a general idea of skewness with particular ways of defining it. (I won't mention kurtosis.) $\endgroup$ – Nick Cox Apr 11 '16 at 15:49
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    $\begingroup$ The historical details here are very interesting to me. My own attempt at a miniature review emphasises that moment-based skewness predates Pearson, although Pearson was mostly more concerned with measuring skewness relative to the mode, as Yule's comments reflect. See stata-journal.com/sjpdf.html?articlenum=st0204 (Indeed, Pearson was obfuscatory in his acknowledgement of prior work on the moment-based measure.) $\endgroup$ – Nick Cox Apr 11 '16 at 17:37
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    $\begingroup$ The extract from Yule helps us see past the extraneous details to the essence of the answer: a distribution in which the positive tail is deemed to be "longer" than the negative tail has positive skewness. Everything else comes down to how one determines the tails and measures their lengths. $\endgroup$ – whuber Apr 11 '16 at 17:42
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    $\begingroup$ I don't see how the answer would lose anything by mentioning one or two other measures of skewness (such as the median-skewness / second Pearson skewness measure) and pointing out that the discussion carries over (just as Nick suggests). $\endgroup$ – Glen_b Apr 11 '16 at 17:44

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