Help with intuition about Chi-Squared distribution and its relation to Normal Distribution

Question

At $\alpha = 0.05$, the significance cutoff of a Chi-squared distribution with 1 df is approximately 3.84, and that if we take the square root of 3.84, it's approximately 1.96, which is the 97.5 percentile of a N(0,1) normal distribution. I understand why this is the case for 1 df, as the Chi-squared distribution is nothing more than a squared N(0,1) distribution.

I also understand that for 2 dfs the Chi-squared distribution is just follows the distribution of the expected sum of two squared random samples from N(0,1), and has a significance threshold of approximately 6.0. Following the logic of the case with 1 df, in the case of 2 dfs we can also be expressed the threshold as the 97.5 percentile of a N(0, ~1.25) Normal distribution, which is about the square-root of 6.0.

Is there a general equation we can use to express a Chi-squared significance threshold (at say alpha = 0.05) for an arbitrary degrees of freedom (x), in terms of a Normal distribution with mean = 0 and standard deviation = y? I was only able to solve the above example for 2 dfs (i.e. x=2, y=1.25) though testing, but if there is some way to generalize this translation between Chi-squared and Normal, I think it would help me to gain more insights into the Chi-squared.

Answer 1

Following the logic of the case with 1 df, in the case of 2 dfs we can also be expressed the threshold as the 97.5 percentile of a N(0, ~1.25) Normal distribution, which is about the square-root of 6.0.

In the 1 df case it doesn't just match at the 97.5 percentile -- all of the percentiles correspond in a similar way.

While there's definitely a connection in the 1 df case, this is not really relevant for other degrees of freedom. How did you arrive at 1.25 (aside from matching its 97.5 quantile to $\sqrt{6}\,$)?

You could as easily say "well, see it's connected to a logistic distribution" and then compute the corresponding scale parameter for a logistic by matching two quantiles in a similar fashion. That doesn't imply they're related in any meaningful sense.

For the chi-squared(1), the two corresponding curves lie one atop the other, rather than just crossing somewhere.

The sum of two chi-squared(1) variates is exponential with mean 2. You can match any individual quantile to some normal distribution but you can't choose a single standard deviation that will make all the quantiles correspond at the same time like you can with 1.df

e.g. The 75th percentile of a chi-squared(2) is about $1.67^2$; to get the 87.5 percentile of a zero-mean normal to be 1.67 you need $\sigma$ to be about $1.448$. Would you want to use a different $\sigma$ for each quantile? I don't see much understanding to be gleaned from that activity.