Compare the difference of two probabilities or a ratio of probabilities? In an experiment with continuous-value random variable, e.g. measuring length of cucumbers. I would like to compare the probability of getting a particular length range in two different conditions. e.g. $P1=P(0.1<L<0.2, {\rm fertilizer\ A})$, $P2=P(0.1<L<0.2, {\rm fertilizer\ B})$.
So I create 2 histograms for the lengths corresponding to each of the 2 fertilizers, $H_A$ and $H_B$. From this histograms, probabilities over ranges of L are calculated from bin counts, etc.
My aim is to show that only for some ranges (or even 1 range) of L the probabilities (or counts of the above histogram) differ while in most other length ranges the probabilities do not differ. 


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*what is a good way to express this difference? The usual ratio $\frac{P1}{P2}$ suffers when $P1$ or $P2$ is zero. a good one is $P1-P2$ but then I would like this to be normalised over P so as to have something like a percentage change for comparison, I thought something along: $\frac{P1-P2}{P1+P2}$

*I would like to plot these probability differences for various L and if possible to use a statistical test which tests the significance of probability differences for just one range and not overall (what a t-test does). I guess the significance of difference depends also on the counts for that particular range of lengths.
 A: Are these distributions sufficiently normal?  You could check a qq-plot to see if they're good enough for your satisfaction.  If so, the area under the fitted normal distribution will never be exactly 0 as it is in a finite sample, so that would be one way to address that issue.  
On another issue, I would take the proportions within given ranges and convert them into the odds of a cucumber falling within that range given that it was grown with that fertilizer.  Then I would use the odds ratio to compare the two.  I think this will be a better approach than using the ratio of probabilities.  
One final note, if the distributions for the two fertilizers differ, then realistically the probabilities of being within a given range couldn't be exactly identical, and so testing a given range for 'significance' doesn't make a lot of sense to me.  I would just do a t-test on the two distributions themselves (I should think Levene's test would also suffice for your purpose).  Having shown that the distributions differ, that means the proportion within a given range will differ, and you could represent the magnitude of the difference for the range that you care about with an odds ratio.
