# rule of thumb for the number of degrees of freedom of a chi-squared to converge to normal

It is said that if $X\sim\chi^2_{(k)}$, then $Y=\frac{X-k}{\sqrt{2k}}$ converges to $Y\sim N(0,1)$ when $k$ tends to $\infty$.

Is there a commonly used rule of thumb for a $k$ to be "big enough" to consider $Y$ standard normal? I'm looking for something analogous to the degrees of freedom greater than 30 for Student's t distribution

• It ought to depend on what you are doing with $Y$, shouldn't it? What application(s) do you have in mind? Note, too, that there are many simple normalizing transformations: for instance, the cube root of $Y$ will be very close to Normal even for fairly small $k$. See Johnson & Kotz's encyclopedic work on distributions for details and other approximations. – whuber Apr 21 '17 at 19:02
• @whuber in a sense, of course it depends. However the t_30 rule of thumb has a broad field of application. I was just wondering if there was something alike. Thanks a lot for the cube-root hint, BTW! – Rafael Apr 21 '17 at 20:01
• The 30 rule of thumb indeed has a broad field of application--and almost as broad a field of misapplication! – whuber Apr 21 '17 at 21:12

I find that to be a pretty plausible explanation for the $n \ge 30$ t distribution rule. One interesting property of the t dist with 30 df is that the most its pdf ever differs from that of the normal is .0045 units. If you want to match that closeness criterion with chi-squared approximation, you'd have to set $k=3425$. You could choose other closeness-criteria and get a much different answer. From a practical standpoint, it's worth noting that the chi-squared approximation maintains quite a bit of skewness - even for fairly large $k$. For instance, when $k=30$, the mode is $-.26$.
• I guess I could have replaced PDF with CDF (that has the effect of taking $k$ down to just $1300$) but I'm not sure that the connection to hypothesis testing is hugely important – jjet Apr 21 '17 at 21:25