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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

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    $\begingroup$ 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. $\endgroup$ – whuber♦ Apr 21 '17 at 19:02
  • $\begingroup$ @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! $\endgroup$ – Rafael Apr 21 '17 at 20:01
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    $\begingroup$ The 30 rule of thumb indeed has a broad field of application--and almost as broad a field of misapplication! $\endgroup$ – whuber♦ Apr 21 '17 at 21:12
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The first website that comes up from your Google search has a pretty interesting take, "In the old days (B.C: before computers) when calculations were done by hand, analysts would use the normal distribution if the degrees of freedom were greater than 30 (for 30 df, the proper multiplier is 2.04; for 60 df, it's 2.00). Otherwise, the t distribution was used. This says as much about the availability of tables of the t distribution as anything else."

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 $-.26$.

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  • $\begingroup$ You ought to be comparing the CDFs and CCDFs: differences in PDFs do not relate directly to decisions made with hypothesis tests. $\endgroup$ – whuber♦ Apr 21 '17 at 21:13
  • $\begingroup$ 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 $\endgroup$ – jjet Apr 21 '17 at 21:25
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    $\begingroup$ If we're not talking about using these distributions about hypothesis testing, then exactly what applications do you have in mind? $\endgroup$ – whuber♦ Apr 21 '17 at 22:29

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