One tends to come across these distributions in one form or the other in data analysis. However till this date, I had never thought through the interconnections between them.

I was aware of them and my knowledge was like independent silos.

  • ChiSquare is square of n standard normal variable with mean = n and variance = 2*n. It is a one parameter distribution
  • F distribution mostly used in analysis of variance where F statistic = ratio of two chi square variables with n1 and n2 degrees of freedom respectively.
  • t distribution used instead of normal when sample size < 30

Here is what I learnt today.

t distribution with n degrees of freedom is nothing but a ratio of standard normal variable to square root of a chi-square variable meaning
t = z*sqrt(n) / y where z is N(0,1) and y is Chisq(n)

The above stuff is used in setting up probabilistic bands for parameter estimates in a regression equation. The fact that the population sigma which is standard normal , turns in to a t statistic when the sample sigma replaces population sigma is pretty good insight for me.Had never thought through this clearly before.

Now whats the connection to F.
t^2 = (z^2/1) / (y^2/n) .. You see that t square statistic is nothing but F(1,n) as z^2 has one degree of freedom.