# What is a basic f-test actually telling me?

When I do an F test(think of an F test you'd learn about in a first year stats paper) I get a ratio of variance between/variance within.

I understand what the indivudual components are i.e the variance between and the variance within. However, I dont understand why they are used as a ratio, why are they compared to one another?

So if I got thefollowing from my F test , 120/20, it seems the F statistic is saying for every 6 units of variance between there is 1 unit of variation within...and it is this that I dont understand. What does it actually mean, or how can it be thought of intuitively.

thanks

If the null hypothesis is true, the two variance estimates should be estimating the same thing ($\sigma^2$), but because of random variation, even when the null hypothesis is true the ratio can be some distance from 1.
e.g. in the case of one-way ANOVA, if the means differ, this inflates the "between" estimate of variance to have expected value larger than $\sigma^2$ (while the within estimate isn't affected, it still has expectation $\sigma^2$). This drives the distribution of the ratio upward (it's no longer from the usual central F-distribution that we have under the null, but - if the assumptions were otherwise true - it would have a noncentral F distribution).
Depending on sample sizes, 120/20 might just be the ratio of two very noisy estimates of the same $\sigma^2$, or the numerator might be estimating something much bigger than $\sigma^2$ (if the df residual is large enough*, the first explanation becomes untenable for F that big.)