Let's say I have $n$ data points with just one categorical feature $x$ and a continuous target variable $y$. I want to divide the possible values of $x$ into subsets such that the value of $y$ doesn't vary much within a subset.
As an example, suppose there are only $5$ possible values of $x$: $x_1,\ldots,x_5$. We observe that the values of $y$ don't vary much across $x_1$ and $x_2$. $y$ doesn't vary much across $x_3$ and $x_5$, but is different from the typical $y$ value for $x_1$ and $x_2$. For $x_4$, $y$ values are quite different from the above groups. So we can say that $(x_1,x_2), (x_3,x_5)$ and $(x_4)$ can be considered as "clusters".
Now what's a concrete way of saying that $y$ doesn't "vary much across $x_1$ and $x_2$"? One natural way to define that is for $y$ to have the same distribution for $x_1$ and $x_2$, i.e. the same mean and standard deviation of $y$ values for both $x_1$ and $x_2$.
Is there a better way to characterize the fact that $y$ doesn't "vary much across $x_1$ and $x_2$? Maybe the way I defined it above is too idealistic and I need a criterion more suited to a real-life dataset?
Are there any popular existing methods or library functions to solve such kind of a problem (i.e. clustering categorical feature values on the basis of continuous target values)?