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Suppose I observe a single pair of numbers $(x, y)$. How do I develop a principled approach to decide whether $x$ is significantly larger than $y$ or vice versa? There are some noise in the measurements, so I kind of have three categories:

  • $x$ is so much larger than $y$, that the uncertainty related to noise is negligible

  • $y$ is so much larger than $x$, that the uncertainty related to noise is negligible

  • $x$ and $y$ are not so different, that it would be possible to make a definite statement

I think statistics would have something to offer here.

First, I have been trying to think what is "significantly larger". I could perhaps formulate a null hypothesis

$$H_{0}: \textrm{$x$ and $y$ were drawn from the same distribution}.$$

However, I don't know how to actually test that. (I guess I can't assume any particular distribution (at least yet), so I would be looking for some general intuition. If it would help, we could perhaps assume normal or uniform distributions.)

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  • $\begingroup$ If you were willing to assume something about that common distribution (such as it is some Normal distribution--but without knowing or assuming anything about its parameters) then tests exist--even when only a single pair $(x,y)$ is observed (which most people find counterintuitive). Without some kind of assumption, though, your task is hopeless, because you have no information about how much of a difference can be considered "negligible." $\endgroup$
    – whuber ♦
    Sep 21, 2015 at 14:31

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If you have no other knowledge and are not willing to make any assumptions, there is nothing you can do.

If you know or are willing to assume that the numbers come from known distributions (e.g. "Normal with mean $\mu$ and sd $\sigma $" or "Poisson with mean $\lambda$" or some such) then there are things you can do, but you'd have to tell us what the distributions are.

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