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Here are some results I found in the literature. I'll assume without loss of generality that $EX=0$ and $EX^2=1$ (in particular, $Skew(X)=EX^3$). I'll also assume that $X$ has a density function $p$.

1. If $p(x)-p(-x)$ changes sign exactly once in $x$, then $$Skew>0 \iff Mean>Median$$ [Mac]. This includes all distributions from the Pearson family as well as lognormal and inverse Gaussian.

2. If $X_1, X_2, \dots$ are independent copies of $X$ and $S_n$ is the sum of the first $n$ copies, then $$\lim_{n\to\infty} Median(S_n) = \frac{-1}{6}Skew(X)$$ This has certain technical assumptions which are not too stringent and can be founded in the referenced paper. (Due to [Hall], with an earlier version proved by [Hald]). Since $$Skew(S_n)>0 \iff Skew(X)>0$$ we thus expect $$Skew(S_n)>0 \iff Median(S_n)<0$$ for sufficiently large n.

The paper actually proves a stronger statement: namely that the median goes as $-{\frac 1 6}k_3+{\frac C n}$, to leading order in $n$. Here $C$ is a certain explicit expression that involves the first five moments of $X$. This allows us to estimate a finite $n$ value for which the skew-median relation will hold. Namely, it holds for $S_n$ provided that $-{\frac 1 6}|k_3|+{\frac {sign(skew)C} n}<0$.

[Hall] P. Hall, On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables, Annals of Probability, 1980,

[Hald] J Haldane,The mode and median of nearly normal distribution with given cumulants, Biometrika, 1942.

[Mac] H MacGillivray, The mean, median, and mode inequality and skewness for a class of densities, Australian Journal of Statistics, 1982.

Here are some results I found in the literature. I'll assume without loss of generality that $EX=0$ and $EX^2=1$ (in particular, $Skew(X)=EX^3$). I'll also assume that $X$ has a density function $p$.

1. If $p(x)-p(-x)$ changes sign exactly once in $x$, then $$Skew>0 \iff Mean>Median$$ [Mac]. This includes all distributions from the Pearson family as well as lognormal and inverse Gaussian.

2. If $X_1, X_2, \dots$ are independent copies of $X$ and $S_n$ is the sum of the first $n$ copies, then $$\lim_{n\to\infty} Median(S_n) = \frac{-1}{6}Skew(X)$$ This has certain technical assumptions which are not too stringent and can be founded in the referenced paper. (Due to [Hall], with an earlier version proved by [Hald]). Since $$Skew(S_n)>0 \iff Skew(X)>0$$ we thus expect $$Skew(S_n)>0 \iff Median(S_n)<0$$ for sufficiently large n.

The paper actually proves a stronger statement: namely that the median goes as $-{\frac 1 6}skew+{\frac C n}$, to leading order in $n$. Here $C$ is a certain explicit expression that involves the first five moments of $X$. This allows us to estimate a finite $n$ value for which the skew-median relation will hold. Namely, it holds for $S_n$ provided that $-{\frac 1 6}|skew|+{\frac {sign(skew)C} n}<0$.

[Hall] P. Hall, On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables, Annals of Probability, 1980,

[Hald] J Haldane,The mode and median of nearly normal distribution with given cumulants, Biometrika, 1942.

[Mac] H MacGillivray, The mean, median, and mode inequality and skewness for a class of densities, Australian Journal of Statistics, 1982.

Here are some results I found in the literature. I'll assume without loss of generality that $EX=0$ and $EX^2=1$ (in particular, $Skew(X)=EX^3$). I'll also assume that $X$ has a density function $p$.

1. If $p(x)-p(-x)$ changes sign exactly once in $x$, then $$Skew>0 \iff Mean>Median$$ [Mac]. This includes all distributions from the Pearson family as well as lognormal and inverse Gaussian.

2. If $X_1, X_2, \dots$ are independent copies of $X$ and $S_n$ is the sum of the first $n$ copies, then $$\lim_{n\to\infty} Median(S_n) = \frac{-1}{6}Skew(X)$$ This has certain technical assumptions which are not too stringent and can be founded in the referenced paper. (Due to [Hall], with an earlier version proved by [Hald]). Since $$Skew(S_n)>0 \iff Skew(X)>0$$ we thus expect $$Skew(S_n)>0 \iff Median(S_n)<0$$ for sufficiently large n.

References:

[Hall] P. Hall, On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables, Annals of Probability, 1980,

[Hald] J Haldane,The mode and median of nearly normal distribution with given cumulants, Biometrika, 1942.

[Mac] H MacGillivray, The mean, median, and mode inequality and skewness for a class of densities, Australian Journal of Statistics, 1982.

Here are some results I found in the literature. I'll assume without loss of generality that $EX=0$ and $EX^2=1$ (in particular, $Skew(X)=EX^3$). I'll also assume that $X$ has a density function $p$.

1. If $p(x)-p(-x)$ changes sign exactly once in $x$, then $$Skew>0 \iff Mean>Median$$ [Mac]. This includes all distributions from the Pearson family as well as lognormal and inverse Gaussian.

2. If $X_1, X_2, \dots$ are independent copies of $X$ and $S_n$ is the sum of the first $n$ copies, then $$\lim_{n\to\infty} Median(S_n) = \frac{-1}{6}Skew(X)$$ This has certain technical assumptions which are not too stringent and can be founded in the referenced paper. (Due to [Hall], with an earlier version proved by [Hald]). Since $$Skew(S_n)>0 \iff Skew(X)>0$$ we thus expect $$Skew(S_n)>0 \iff Median(S_n)<0$$ for sufficiently large n.

The paper actually proves a stronger statement: namely that the median goes as $-{\frac 1 6}k_3+{\frac C n}$, to leading order in $n$. Here $C$ is a certain explicit expression that involves the first five moments of $X$. This allows us to estimate a finite $n$ value for which the skew-median relation will hold. Namely, it holds for $S_n$ provided that $-{\frac 1 6}|k_3|+{\frac {sign(skew)C} n}<0$.

[Hall] P. Hall, On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables, Annals of Probability, 1980,

[Hald] J Haldane,The mode and median of nearly normal distribution with given cumulants, Biometrika, 1942.

[Mac] H MacGillivray, The mean, median, and mode inequality and skewness for a class of densities, Australian Journal of Statistics, 1982.

Here are some results I found in the literature. I'll assume without loss of generality that $EX=0$ and $EX^2=1$ (in particular, $Skew(X)=EX^3$). I'll also assume that $X$ has a density function $p$.

1. If $p(x)-p(-x)$ changes sign exactly once in $x$, then $Sign(Skew)=Sign(Mean-Median)$ [Mac]. This includes all distributions from the Pearson family as well as lognormal and inverse Gaussian.

2. If $X_1, \dots, X_n$ are independent copies of $X$ and $S_n$ is their sum, then $Median(S_n)\to -Skew(X_1)/6$. There are certain other technical assumptions which are not too stringent and can be founded in the referenced paper. (Due to [Hall], with an earlier version proved by [Hald]). Since $Sign(Skew(S_n))=Sign(Skew(X_1))$ we thus expect $Sign(Skew(S_n))=-Sign(Median(S_n))$ for sufficiently large n.

References:

[Hall] P. Hall, On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables, Annals of Probability, 1980,

[Hald] J Haldane,The mode and median of nearly normal distribution with given cumulants, Biometrika, 1942.

[Mac] H MacGillivray, The mean, median, and mode inequality and skewness for a class of densities, Australian Journal of Statistics, 1982.

Here are some results I found in the literature. I'll assume without loss of generality that $EX=0$ and $EX^2=1$ (in particular, $Skew(X)=EX^3$). I'll also assume that $X$ has a density function $p$.

1. If $p(x)-p(-x)$ changes sign exactly once in $x$, then $$Skew>0 \iff Mean>Median$$ [Mac]. This includes all distributions from the Pearson family as well as lognormal and inverse Gaussian.

2. If $X_1, X_2, \dots$ are independent copies of $X$ and $S_n$ is the sum of the first $n$ copies, then $$\lim_{n\to\infty} Median(S_n) = \frac{-1}{6}Skew(X)$$ This has certain technical assumptions which are not too stringent and can be founded in the referenced paper. (Due to [Hall], with an earlier version proved by [Hald]). Since $$Skew(S_n)>0 \iff Skew(X)>0$$ we thus expect $$Skew(S_n)>0 \iff Median(S_n)<0$$ for sufficiently large n.

References:

[Hall] P. Hall, On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables, Annals of Probability, 1980,

[Hald] J Haldane,The mode and median of nearly normal distribution with given cumulants, Biometrika, 1942.

[Mac] H MacGillivray, The mean, median, and mode inequality and skewness for a class of densities, Australian Journal of Statistics, 1982.