Chi square is useful - but why square?

[antevante]

Hello!
While conducting chi square -tests I asked my teacher what the "square" means, and why it was there? Why chi squared, and not just chi?
He couldn't give an answer so he told me to find out, as a homework :surprise: ... After a quarter of "Googeling" I gave up.
So, is there anyone out there who knows why there is a square in chi square??

 

[uart]

The "chi square" is just a name. The chi square function is actually a particular member of the Gamma(a,b) family of distributions. These are all of the form x^(a-1) exp(-x/b) with appropriate normalization to make them valid probability density functions.

If you start with some number r of independent Normal(0,1) random variables and then form a new random variable by adding the squared value of each of these r variables together then this new variable has a Chi Square distribution. This is the Chi Squared functions claim to fame and obviously makes it useful in determining the distribution of a sample variance. The "squared" in the name is nothing more than a reminder that it is the distribution of a sum of squares.

 

[ahrkron]

...or maybe you are wondering why we want the sum of squares?

There are two ways to answer this: the very simple-minded explanation is that, by using the squares, you don't get cancellation between deviations, that would otherwise give you a small deviation in cases where your model is not really good.

Another way o answering this comes from the fact that in many cases, you assume your distribution of errors to be gaussian, and then the sqared deviation shows itself in the exponent of the PDF.

 

[Jin314159]

From my understanding, the Chi-Square distribution is the standard normal distribution squared.

 

[uart]

From my understanding, the Chi-Square distribution is the standard normal distribution squared.

Well that actually is one particular member of the Chi-Square family (for one degree of freedom). The general Chi-Squared of "r" degrees of freedom is the distribution of the sum or "r" independent standard normal random variables.

 

[Jin314159]

Well that actually is one particular member of the Chi-Square family (for one degree of freedom). The general Chi-Squared of "r" degrees of freedom is the distribution of the sum or "r" independent standard normal random variables.

Oh yea... there's like a whole family of Chi-squares with varying degrees of freedom.

 

[uart]

From my understanding, the Chi-Square distribution is the standard normal distribution squared.

BTW, there was one thing I missed in your statement before. I know the following may sound a bit “nit picky” but it is a very important distinction.

There is a big difference between “the standard normal distribution squared” and the distribution of the new random variable that is obtained by taking a standard normal random variable and squaring it.

A standard normal distribution squared would simply be of the form,
$$\exp(-\frac{x^2}{2} ) * \exp(-\frac{x^2}{2}) = \exp(-x^2)$$

On the other hand the distribution of the square of a standard normal random variable can be shown to be of the form, $$x^{-1/2} \exp(-x/2)$$, which is the 1st order Chi-Squared function (after appropriate normalization of course).