Does a Chi-Square random variable $\chi^2_1$ mean that only one normal random variable was taken? I'm trying to understand how Chi-Square variables work.
So far, I know that a Chi-Square random variable, $\chi^2$, means that one random value has been taken from a normally distributed graph. Let's say it was the standard normal distribution. This means, $\chi^2$ has a high probability of being zero or near zero.
Here's what I don't understand; How many degrees of freedom does $\chi^2$ have? If it only represents one random variable, then it has zero degrees of freedom, doesn't it?
For example, if I take 5 random variables from a normal distribution, is it $\chi^2_1+\chi^2_1+\chi^2_1+\chi^2_1+\chi^2_1= \chi^2_5$ or is one somehow not counted?

If $X_1, \ldots, X_n$ are independent standard normal random variables, then $X_1^2 + \ldots + X_n^2$ has a ChiSquare distribution with $n$ degrees of freedom.
