# What sample statistics is increasing (or non-decreasing) as data grows

First please forgive me of using this non-mathematical terminology in the title.

Assume we have two series of real numbers $X_1, X_2, \ldots, X_M$ and $Y_1, Y_2, \ldots, Y_N$ where it is possible for $X_i=X_j$ and $i \neq j$. Now if we combine series $X$ and $Y$, the resulting set of real numbers $X_1, X_2, \ldots, X_M, Y_1, Y_2, \ldots, Y_N$ is denoted as $X+Y$. Finally let us assume a statistics $T$ can be defined for $X$, $Y$, and $X+Y$ similarly.

My question is: what statistics is guaranteed to have the property: $T(X+Y) >= T(X)$?

For example, suppose we define $T(X)=max(X_1, X_2, \ldots, X_M)$. It follows that $T(X+Y)=max(X_1, X_2, \ldots, X_M, Y_1, Y_2, \ldots, Y_N ) \geq max(X_1, X_2, \ldots, X_M) = T(X)$.

So the maximum statistics satisfy this property. Another example would be $T(X)=card(X)$. In both cases, as data grows (from $X$ to $X+Y$), the statistics is monotonically non-decreasing.

On the other hand, the median statistics does not necessarily satisfy the property.

My question is, in general, what types of statistics would hold the property of $T(X+Y) >= T(X)$? I am particular in range statistics.

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Although this question looks focused, it really is extremely broad due to the huge disparate collection of solutions available. So that the question can be narrowed and made effectively answerable, could you please indicate what the immediate problem is that you are attempting to solve? – whuber Mar 21 '12 at 18:38
Suppose we are measuring the weight of a pile of apples ($X$) and a pile of oranges ($Y$). When I mix apples and oranges together, my data becomes heterogenous. My conjecture is that this heterogeneity will lead to wider distribution (not necessarily true!) and I would expect some statistics to reflect that. – Haining Yu Mar 21 '12 at 19:10
OK, at a minimum then it appears you are looking at statistics that are invariant under permutation and you are really seeking measures of variability. There are still too many possible answers to choose from and we still don't have any criteria to help you decide which will work well in your application. Exactly what do you intend to do with such a statistic? What inferences or decisions will you make? What is the problem you really face here? – whuber Mar 21 '12 at 19:15
For some reason my previous comment was not complete. Please ignore it and I will try to post again and delete it if I can. Thank you for your patience. – Haining Yu Mar 21 '12 at 19:19
Your comments are useful. In the latest, you introduce two distinct ideas in a way that falsely suggests they are the same. "Guessing" a weight is, more generally, estimating a parameter. However, "heterogeneity" is a property of the data. The heterogeneity is partly independent of the variability of the estimator. It is possible for the heterogeneity to grow while the estimator gets more and more precise. So, where does your interest lie: in characterizing a population from which you are sampling or in characterizing the samples themselves? – whuber Mar 21 '12 at 19:37