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Measure for how much a data deviates from "perfectly symmetric/ideally distributed" in data's units?

How do I produce a skewness/asymmetry/something measure that's in the same units as the data may be in, e.g. metres?

So if I e.g. want to know "how many metres is this metric data 'skewed' to left/right" (compared to if it was perfectly symmetric), then what's such measure?

PCA may be able to produce one such measure by comparing the first PC to a perfectly straight line. This would approximate e.g. how much the possible "top" of the data is "off" from perfectly symmetric data. But PCA does not describe how the asymmetry is distributed, whereas https://en.wikipedia.org/wiki/Skewness does. But skewness does not give a deviation in e.g. metres does it?

Product of inertia is also possibly related.

mavavilj
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    how many metres is this data skewed to left/right compared to what? – user2974951 Jun 29 '21 at 10:18
  • @user2974951 Compared to perfectly symmetric. – mavavilj Jun 29 '21 at 10:21
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    Skewness measures have no units. You need other standards for comparison. Note that skewness based on third and second moments is far from the only summary measure – Nick Cox Jun 29 '21 at 10:33
  • @NickCox "other standards", like? – mavavilj Jun 29 '21 at 10:37
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    Like what you expect for distributions that might apply. Something with length dimensions might be expected to be positively skewed and/or follow a gamma or lognormal, for example. You could tell us more if you have particular datasets in mind. – Nick Cox Jun 29 '21 at 11:12
  • @NickCox Is it not possible to take e.g. the difference between the new mean and mode? I.e. in perfectly symmetric mean is in the middle, then in skewed versions we could measure how much mode has shifted from mean? https://en.wikipedia.org/wiki/Skewness#/media/File:Relationship_between_mean_and_median_under_different_skewness.png – mavavilj Jun 29 '21 at 18:44
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    Yes, The difference between mean and mode clearly has the units of the variable being analysed. That is the basis of one skewness measure, once much used, suggested by Karl Pearson, but the difficulty is estimating the mode reproducibly. – Nick Cox Jun 29 '21 at 19:54
  • https://stats.stackexchange.com/questions/145159 is relevant and might answer the questions that must be lurking beneath this one. The problem with the present formulation is that it refers too vaguely and generally to "a skewness measure." Some such measures (such as a standardized third moment) are unitless while others (such as differences among means, medians, and modes) are in the original units of the data; perhaps others (of which I am less familiar) are in other units still. That leaves this question in need of clarification. – whuber Jun 29 '21 at 20:01
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    Re the edits: It depends on what you mean by "skewness." At least two different concepts are in common use. One is that it tells us something about the shape of a distribution. As such, it is necessarily unitless. The other is grounded in various proposals to measure "skewness," such as the difference between mean and median. But those obviously do not permit comparing skewnesses among different kinds of data and depend on the units of measurement, which makes them rather specialized and of limited use. That's why we need you to tell us what "skewness" is supposed to measure. – whuber Jun 30 '21 at 17:02
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    Re your comments about PCA: This is a full second-order description of the shape of a multivariate distribution. (The first-order description consists of the expectations of the variables). No standard PCA output tells you anything about asymmetry. For a general approach to assessing asymmetry, see https://stats.stackexchange.com/a/145366/919. – whuber Jun 30 '21 at 17:17
  • To add to Whuber's comment, PCA does tell you things about a distribution, but they are not the things you want to know about in this context. – Galen Jun 30 '21 at 17:19
  • There is a lot of precedence around "symmetry" in a statistical context, as mentioned above, but I would also add that there are opportunities to even further generalize to transformational invariance instead of just reflective symmetry. – Galen Jun 30 '21 at 17:28
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    @2.7182818284590452353602874713 I discuss that generalization at https://stats.stackexchange.com/a/29010/919. – whuber Feb 05 '22 at 16:12

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