TITLE: Example of a Distribution that is not symmetric, but has a skewness of zero.
QUESTION [5 upvotes]: In my probably class we saw that if a distribution is symmetric then the skewness will be zero. Intuitively this makes sense to me since if a data set is symmetric than for each point that is distance 'd' above the mean there will be a point that is distance 'd' below the mean (Although, in practice this is probably just very close to zero and actually zero).
However, it was implied in class that the converse is not true. That is, there may exist a distribution which is not symmetric, or skewed, but the skewness is zero.
Can anyone give me an actual example of a distribution that is skewed, but the values of the skewness equals zero?
Cheers

REPLY [3 votes]: We can construct a simple example such that $E(X)=0$ and $E(X^3)=0$, which implies that the third central moment is 0 and thus zero skewness. This gives us a hint. For instance, we only need take $X\in\{-3,-1,2\}$ and $P(X=-3)=a, P(X=-1)=b, P(X=2)=1-a-b$. Now, we have two unknows and two equations,
\begin{equation}
(-3)*a+(-1)*b+2*(1-a-b)=0~~\textrm{and}~~(-3)^3*a+(-1)^3*b+2^3(1-a-b)=0
\end{equation}
By solving $a=1/10$ and $b=1/2$, we get an asymmetric distribution and the skewness is 0. It is asymmetric by construction.