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If I look at a quantile plot comparing normal distribution and other data, and some of the points lie below or above the line, how do I know what that represents? What I am asking is - how do I know if the data set has a left or right skew?

kjetil b halvorsen
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Luc
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    Do you want to edit your post to include an example plot so someone can give you a concrete response? – mdewey Jan 31 '17 at 09:03
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    Are you asking about a [quantile-quantile plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot) or some other plot involving quantiles? If so, does [this question](http://stats.stackexchange.com/questions/101274/how-to-interpret-a-qq-plot) solve your problem? – Glen_b Jan 31 '17 at 10:46
  • See also http://stats.stackexchange.com/questions/145159/how-to-tell-if-my-data-distribution-is-symmetric – Nick Cox Jan 31 '17 at 11:34
  • @NickCox At the end of the sequence of links for the link you give is a [program](https://xiongge.shinyapps.io/QQplots/) that only modifies skewness and kurtosis (called tailedness by the author). This does not answer the question as it leaves out all higher moments AND the role of outliers. – Carl Feb 01 '17 at 04:40
  • @Carl The thread i cited is I think highly relevant to this one. If I understand you correctly, you're dissenting from one answer by a third party in that thread. The better place to comment is surely there. – Nick Cox Feb 01 '17 at 09:16
  • @NickCox The last time copied an answer to another question, the moderator did so as well. That created two copies of my answer on the same post, and a comment to my attention by the moderator. – Carl Feb 01 '17 at 17:11
  • @Carl I am suggesting a comment, not another answer, but it's entirely your choice what to post. – Nick Cox Feb 01 '17 at 17:13
  • @NickCox Yes, I am dissenting, not to be a 'maverick.' Skewness is peculiar enough that it's truth value is a calculation, not a graph. – Carl Feb 01 '17 at 17:15
  • @NickCox I will, eventually. Thank-you for the comment. – Carl Feb 01 '17 at 17:16
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    I am not anxious to re-open a discussion that has already happened, but I hold that skewness is not a single Platonic property but definable by scalar measures in numerous different ways. That point of view is expounded in the thread I cited. Just two minute examples: a Weibull with moment skewness is not symmetric; there are many binomials with mean = median = mode which are graphically skew. – Nick Cox Feb 01 '17 at 17:20
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    @NickCox Skewness has to be defined before one can reasonably expect any plot to reveal skweness. Pick one. Using the moment definition, one can numerically calculate skewness for distributions that have no defined skewness, e.g. Student's t for [$v\leq 3$](https://en.wikipedia.org/wiki/Student's_t-distribution). One cannot then invert the Q-Q graph and talk about relative tail heaviness for tails that are of random heaviness. – Carl Feb 01 '17 at 17:48
  • Agreed that a Q-Q plot isn't designed to show skewness alone and reading skewness off it needs care: your statement would be stronger. But I can just define skewness as lack of symmetry and then plotting (upper quantile $-$ lower quantile) is informative. I don't have to jump for any single measure before I think about skewness. – Nick Cox Feb 01 '17 at 17:51
  • @NickCox One can Q-Q plot anything. For a Cauchy distribution, using one set of outer quantiles we would conclude that it is heavy left tailed, or heavy right tailed, where it is actually symmetric. To "think" of skewness is not enough, one has to nail it down precisely with a definition. – Carl Feb 01 '17 at 18:12
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    I can only speak for myself with authority. I thought of skewness just using graphs for many years without paying very much attention to ways to define it precisely. Once I had learned of ways to measure skewness, I then had to unlearn the idea, or move towards the further idea that there are many possible measures and one can jump between them according to the problem. – Nick Cox Feb 01 '17 at 18:17
  • @NickCox I think the question above is unique enough that it merits attention. The other questions did not result in extensive discussion, they are different, e.g, symmetry assumption. Also, correlation can be Pearson, Spearman, extended R and not *r,* etc. So can skewness, i.e., it has multiple definitions. To speak of skewness, one has to choose one definition just like for correlation. – Carl Feb 01 '17 at 18:25
  • I think you'd have a hard time persuading the forum to re-open this. What has more chance of lift-off is your posting a question on skewness to which you provide one answer with the expectation that others might add theirs. In fact, you've already written the answer, below! – Nick Cox Feb 01 '17 at 18:38

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Well, you wouldn't know because both tails are effected by skewness, kurtosis and other moments. For example, and although an accepted answer leads to an examination that only modifies skewness and kurtosis (called tailedness by the author). This does not answer the question as it leaves out all higher moments AND the role of outliers upon Q-Q plots. Indeed, the influence of higher moments and outliers is not even discussed. To say this another way, a normal Q-Q plot can be used to examine data that is not normally distributed, and in so doing, the skewness is only indirectly shown. One method of determining skewness is to just calculate it.

Positive skewness has been said to have a longer or fatter right (than left) tail. Fat and long tails are not quite the same things, such that the usual graphical explanation of skewness is somewhat ambiguous. For example, we could have a fat left tail and a long right tail, and that is not uncommon. Thus, the best definition of skewness is from the formula used to calculate it. That is, although we plot what skewness looks like on a histogram or Q-Q plot in particular cases, there is no unique graphic description of skewness, only a mathematically one. This lack of uniqueness prevents us from inverting the problem and determining skewness from a graph, except in cases that have been so narrowly defined, that that graphic interpretation is unique.

Carl
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  • I'm certainly not disagreeing with you, but wouldn't the best way of **seeing** and describing skew be to just look at the distribution? i.e. make a histogram of the data? – IWS Jan 31 '17 at 08:05
  • @IWS Histograms are often the first thing we look at, but the OP didn't ask about them. Histograms can suggest skewness, but it is only the calculation of skewness that is definitive for skewness. It is a tautology; skewness(of distribution)=skewness(calculated). – Carl Jan 31 '17 at 16:45
  • @whuber If this is closed as duplicate, would you then be so kind as to move my answer, as well. That is, if you do not think it incorrect, please. – Carl Feb 01 '17 at 05:07