# How to find a critical value of a distribution ?

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

## Answers and Replies

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:
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

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

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!!

Homework Helper
MHB
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!!

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.

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.