Where does the Chi-square equation come from? A Chi-Square random variable is defined as $Χ^2_{df} = \frac{( n - 1 )  s^2 }{σ^2}$.
I cannot find a source for this anywhere on the internet, and my textbook is of no help. 
Does anyone know where this formula came from and why it makes up what we call the Chi-Square?
 A: Chi square distribution with $n$ degrees of freedom is the sum of $n$ independent normal distributions.
If $X_i \sim N(0,1) \ ;i=1,\ldots,n$ then $\sum_{i=1}^n{X_i}^2 \sim \chi_n^2$.
It is mainly used for Testing of Hypothesis and regression statistics.
What you are seeing is another way of writing Chi-square. It was introduced to us in a similar manner in probability.
Consider $X_i \sim N(0,\sigma^2 )$. Then $\frac {X_i^2}{\sigma^2} \sim \Gamma (\frac 12,\frac 12)$. Adding we get, $\sum_{i=1}^n \frac {X_i^2}{\sigma^2} \sim \Gamma (\frac n2,\frac 12)$ whic is precisely Chi square distribution.
For more general result,please check Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$ 
I don't know how the formula originated, but can give you one of major (probably) reasons behind  use of this formula. Suppose you want to find $E(s^2)$, then you can use this formula. This leads to chi square test.
Also I think that it is worth mentioning that chi squared is the distribution of Quadratic forms for standard normal.
