Dear Rick Wicklin,

You ask if anybody knows where the approximation ≈ d(1-2/(9d))3 comes from. I looked up chi-square in the bible of functions, Abramowitz and Stegun, and stumbled upon formula 26.4.14, which states that if d>30, then the transformed variable x_2 = ( (x/d)^(1/3) –(1-2/9d) ) / \sqrt(2/9d) is standard normal distributed. And when that’s the case, the median Is X_2=0 and hence the above approximation.

This approximation is a statement about the entire distribution; however from your graphs it seems safe to apply the median approximation for even lower degrees of freedom.

(Where Abramowitz and Stegun have their result from I have no idea.)