# How to find a critical value of a distribution ?

1. Nov 18, 2011

### szandara

I am trying to find an ANALYTICAL way to extract a critical value from a generic distribution. I need a way to find it analytically because I am trying to find a relationship between the critical values.

I always find methods that depends on the distribution tables, but I would like to know how these tables are built.

I did not find anything on the internet?

any advice? ( in particular I am working with Gaussian and ChiSquare distributions)

thanks

Simone

2. Nov 18, 2011

### Stephen Tashi

Computing critical values for a statistical test from distributions involves solving equations such as $\int_c^\infty f(x) dx = 0.05$ for the value $c$. If there is no "analytical" solution for writing the integral then there is no "analytical" way to compute the critical value. I am assuming that by "analytical", you mean a relatively simple "closed form" formula. Is that what you mean? I know of no "analytical" way to solve the intergrals involved in the normal and chi-squared distributions. The tables are built by using numerical approximations to solve the equations.

Last edited: Nov 18, 2011
3. Nov 19, 2011

### szandara

yes that was actually what I wanted to know.

So, I must suppose that there's no closed form for Gaussian and Chi Square distributions?
I havent tried myself yet.

S

4. Nov 19, 2011

### chiro

For those problems, that is equivalent to solving the anti-derivative of those functions, which so far as Stephen Tashi has said, has not been found (yet).

If you find it, be sure to let us know!!

5. Nov 19, 2011

### I like Serena

I believe they have been proven not to exist using only a finite expression of standard functions.

However, just like sin(x) is just a conventional name for a function, we also have $\Phi(x)$ which represents the anti-derivative of the standard normal distribution function.

6. Nov 19, 2011

### chiro

Yeah I said the wrong thing, thanks for pointing that out.

I meant to say an analytic way of solving for c, which for many cases (even polynomials with degree higher than say 4 or 5) present a challenge.