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It turns out there is a lower bound on the skewness $g_1$ of any strictly positive set of data having a given mean μ and standard deviation σ: $$ g_1 > \sigma/\mu - \mu/\sigma. $$ Although discussed as a fresh result in some recent literature, it seems to me likely to be quite ancient — for reasons I outlined in this PubPeer post (which also contains an elementary proof).

Coming here to ask this Q, I encountered 2 questions to which this result applied immediately: see my answers here and here. So this suggests the lower bound is at least less well-known than it should be. But could it possibly be of recent vintage?

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    $\begingroup$ You appear to be at the start of this Terence Tao story, my point being that sometimes even relatively basic results exist only on the periphery of the literature: terrytao.wordpress.com/2019/12/03/… $\endgroup$
    – Lukas Lohse
    Commented Jun 29 at 20:31
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    $\begingroup$ In my experience, statistical theory from the late 40s, 50s and 60s has been heavily forgotten, particularly results concerning cumulants. It's possible that this result would have been known at the time, but I haven't encountered it myself before. However, while this result is a nice little one, I guess there is also the question of how relevant it actually is? Perhaps whenever is was rediscovered, it was never shared broadly? It's very possible that it has never been published before, due to the lack of a clear situation to publish it. Perhaps a textbook or class exercise? $\endgroup$
    – Guillaume Dehaene
    Commented Jul 1 at 8:27
  • $\begingroup$ Guillaume, I would argue the result has found a worthwhile application in this PubPeer comment. $\endgroup$
    – David C. Norris
    Commented Jul 5 at 13:06
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    $\begingroup$ What do you mean by vintage? Do you want to know the year that this result first appeared? $\endgroup$
    – Sextus Empiricus
    Commented Jul 7 at 21:06
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    $\begingroup$ Sextus, thanks for requesting this clarification. Yes, ideally I would like to know when the result first appeared in print. But I know 'earliest' is a tall order! I honestly would be happy for any archaeological find that increases the lower bound on the result's age. $\endgroup$
    – David C. Norris
    Commented Jul 8 at 1:14

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$\newcommand{\s}{\sigma} \newcommand{\a}{\mu}$ I agree that this could have been known a century ago. In any case it has a quick proof, following my answer to one of the linked questions.

Let the variable be $X$, with mean $\mu$ and standard deviation $\sigma$.

First, by the Cauchy-Schwarz inequality: \begin{align} \left|E[X^{1/2}\,X^{3/2}\,]\right|^2 &\le E[X]\, E[X^3]\\ (E[X^2])^2 &\le E[X]E[X^3]\\ \end{align} where $X^{1/2}$ and $X^{3/2}$ are well-defined by the positivity of $X$. So $$E[X^3]\ge\frac{E[X^2]^2}{E[X]} =\frac{(\mu^2+\s^2)^2}\mu$$ Then \begin{align} \s^3 \text{skewness} &= E[(X-\a)^3]\\ &= E[X^3]-3\a E[X^2]+3\a^2E[X]-\a^3\\ &=E[X^3]-3\a(\a^2+\s^2)+3\a^3-\a^3\\ &\ge (\a^2+\s^2)^2/\a -3\a\s^2-\a^3\\ &= \s^4/\a - \a\s^2\\ \text{skewness} &\ge \s/\a - \a/\s, \ QED \end{align}

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    $\begingroup$ This is very neat. There remains a different but related question of an analogue inequality for sample measures, as sample skewness and the coefficient of variation are bounded by sample size, as has often been rediscovered. $\endgroup$
    – Nick Cox
    Commented Jul 1 at 14:50
  • $\begingroup$ Happy to award the 50-point bounty to this concise proof, which is a real help to moving forward with this. I'm not accepting it as answer, however, since I want to press forward specifically on the question of vintage. $\endgroup$
    – David C. Norris
    Commented Jul 7 at 7:31

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