# How to find a critical value of a distribution ?

## Main Question or Discussion Point

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

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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.

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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

chiro
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!!

I like Serena
Homework Helper
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.

chiro
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.