Can chi-squared distribution be left-skewed? I'm learning chi-squared distribution and know that as degree of freedom increases, right-skewed chi-squared distribution will approximate to normal distribution shape. I'm wondering can its distribution become left-skewed? How should I approach to think about this question?
Thanks.
 A: No, the skewness of the chi-squared distribution with $k$ degrees-of-freedom is:
$$\mathbb{Skew} = \sqrt{\frac{8}{k}},$$
which is positive for all $k > 0$.  The distribution is asymptotically unskewed as $k \rightarrow \infty$, and indeed, in this case it converges to the normal distribution (in an appropriately standardised sense).
A: No.
Traditionally, chi-squared distributions have integer parameter values, $\nu =1, 2, 3, \dots,$ called degrees of freedom. See Wikipedia for details, and for some plots of density functions--all of them right skewed, with skewness
decreasing as degrees of freedom $\nu$ increase.
However, possibly as a convenience for matching chi-squared distributions with other members of the gamma family of distributions, some software packages are programmed to
give numerical answers for positive values of $\nu$ that
are not integers:  95th percentiles of chi-squared
distributions with $\nu = .5, 1, 1.3, 2$ are about
$2.4202, 3.8414, 4.5448, 5.9915,$ respectively, according to R. [Ordinarily, you will find only the 2nd and 4th of these values in a printed table of chi-squared distributions.]
qchisq(.95, c(.5,1,1.3,2))
[1] 2.420232 3.841459 4.544779 5.991465

Here are plots of chi-squared density function for
$\nu = 1$ [black], $\nu = 2$ [dotted red], and $\nu=2$
[blue].

R code for figure:
hdr = "Right skewed Densities of CHISQ(1), CHIS1(1.5), CHISQ(2)"
curve(dchisq(x,1), 0,5, ylim=c(0,3), lwd=2, ylab="Density", main=hdr, n = 10001)
 curve(dchisq(x,1.5), add=T, lwd=2, lty = "dotted", col="red", n = 1001)
 curve(dchisq(x,2), add=T, lwd=2, col="blue", n = 10001)
  abline(v=0, col="green2")
  abline(h=0, col="green2")

Note: There are generalized chi-squared distributions, which
can be left skewed; see Wikipedia.
