# Test for equality of two numbers

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.)

• 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." – whuber Sep 21 '15 at 14:31

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.