# Is there a good summary statistic that indicates that a vector has a high average _and_ a high sum?

Suppose I have a large number of positive vectors of different lengths that are drawn from a heavy tailed distribution. I'd like to summarize them in a way that ranks the ones with a large sum and a large average over ones that have only a large sum or only a large average. For instance, say I have

A = [10, .1]
B = [0.6, 0.6, 0.6, 0.6, ..., 0.6] (length 20)
C = [.1, .1, .1, ..., .1] (length 1000)

If I rank each by their average, I have A > B > C. If I rank each by its sum, I have C > B > A. But really, B is the most interesting to me, because it has the most big numbers.

I know that in this example, taking the median would give me what I want, but I don't think that's exactly what I'm looking for in general—I can easily construct examples where the median wouldn't give me what I want.

Is this a standard problem? Are there standard statistics that would help me here?

• $(mean)^2+(sum)^2?$
– Dave
Aug 8, 2020 at 23:36
• Note that in fact A has a larger sum (as well as a larger mean) than B so it might be difficult to justify anything which selected B. Also note that the sum is the mean multiplied by the number of terms Aug 9, 2020 at 0:06
• @Henry well, imagine B had length 20
– crf
Aug 9, 2020 at 0:25

Because sum and mean goes together by $$mean = \dfrac{sum}{n}$$, I suppose you can use either sum or mean together with $$n$$. Intuitively you'll want higher sum or mean with lower $$n$$.
Say that there are cases where two vectors have same sum or mean and same $$n$$, and you are interested in which vector has larger values. Median helps here.