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I have often read that there is a huge connection between the normal distribution and several other distributions. But these were only mathematical explanations.

What's the "real" connection between these 3 distributions (normal, $\chi^2$, and F)?

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Don't the answers to your previous question also answer this one? – whuber♦ Aug 27 '12 at 22:25
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The simple answer is: if you square a normal, you get a $\chi^2$ and if you take the ratio of two $\chi^2$'s, you get a variable with an $F$ distribution. But, your quotations around "real" in your question make me suspect you're asking something else. Can you provide some more details? It may help you know the context in which this question arose. – Macro Aug 27 '12 at 22:57
@Macro Not quite right. To get the F you need the chi squares to be independent and each needs to be divided by its degrees of freedom before taking the ratio. – Michael Chernick Aug 27 '12 at 23:13
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Yes thank you for that correction @Michael but the real purpose of my comment was to clarify what the question is really asking - I suspect that the OP is already aware of these facts (maybe hearing them in a class is what motivated the question) and was asking a) whether there is a deeper explanation about the connection between these distributions or b) perhaps a derivation of their relationships or possibly c) something else? I think the quotations around "real" is what made me think a something more than the simple answer was being requested. – Macro Aug 28 '12 at 0:03
@Macro I got that impression too. But what else is there to say. My answer and your comment describe the connection between the three distributions. It is not complicated mathematics. – Michael Chernick Aug 28 '12 at 0:17

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It is simple. Chi square random variables are sums of squared independent standard normal random variables and an F random variable is the ratio of two independent chi square random variables divided by their degrees of freedom. That explains why the F distribution comes about in the analysis of variance. The chi square comes up when estimating the variance of a normal distribution.

Edit: Perhaps this may make it clearer:

$$N_1,...,N_S {\stackrel{\mathrm{iid}}{\sim}} \mathcal{N}(0,1)\ \longrightarrow Y=N_1^2+...+N_s^2 \sim \chi^2_s$$

$$R\sim\chi^2_r \ \ \ {\rm and} \ \ \ S\sim\chi^2_s {\rm (independent)} \longrightarrow Y=\frac{\frac{1}{r}R}{\frac{1}{s}S}\sim F_{r,s}$$

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In the phrase "Chi square random variables are sums of squared standard normal random variables " do the standard normal random variables need to be independent? – Dilip Sarwate Aug 28 '12 at 0:53
@DilipSarwate Yes – Michael Chernick Aug 28 '12 at 1:28
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Then why not say so up front so as to give a completely correct answer from the get go, or edit your answer now that the omission has been pointed out? – Dilip Sarwate Aug 28 '12 at 1:38

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