# What distribution do I get when I square numbers from a normal distribution and add them together?

I have 60 numbers drawn from a normal distribution with mean 0 and standard deviation of 1.

1 realization:

I then take the sum of the 60 values.

I do this 1000 times and plot a histogram of the various sums I get:

And I am able to fit the histogram with a normal distribution with a mean of 0 and a standard deviation of sqrt(60).

So far so good.

I now want to square each of my 60 numbers to get (for example) a realization like this:

If I again take the sum of the 60 values and repeat 1000 times I get a different histogram:

My question: what function do I use to fit this histogram? Is it another normal distribution? Maybe chi-squared? What parameters do I use?

• What is the horizontal axis in your "realization" graphs? Apr 16 at 1:07
• I'm sure those are indices. It really doesn't matter. A point graph would be more appropriate than a line graph. The lines imply some connection between the points but the only connection between them is the negligible connection provided by the pseudo-random number generator. Apr 16 at 19:06

If $$z$$ has a standard normal distribution (mean of zero and variance of one), then $$z^2$$ has a chi-squared distribution with 1 degree of freedom.
Furthermore, the sum of $$x$$ and $$y$$ from chi-squared distributions with degrees freedom $$\nu_x$$ and $$\nu_y$$ is itself chi-squared distributed (with $$\nu_x+\nu_y$$ degrees of freedom).