I have a tuple of variables $$(X_1, ..., X_n)$$ that are a sample of what I call a "benchmark test", eg. each variable represent a value that depends on the performance of a system, but individual variables may be affected or skewed "a little" depending on the kind of test done. All variables are considered of equal importance.
So while each variable is roughly proportional to some extend to the imagined "performance" $p$ of the system, they all have a different distribution, eg. $X_3$ may have a reasonable variance and almost the same mean of about 100 for equal systems, while $X_7$ may be of smaller variance between the systems and has a mean of 1.
Is it possible now to choose a model distribution for $X_i$ and create a meaningful mapping of a single sample, like $$E(x_1, ... , x_n)$$, that is approximately proportional to the "performance" $p$, without knowing any sample, therefore any mean or variance in advance?
A trivial attempt, by just adding $x_1, ...,x_n$ obviously fails, as any $x_7$ drawn from a $X_7$ with small mean would not affect the outcome as much as than a large and more variant $x_3$ drawn from $X_3$.