# What's a linear aggregate?

A paper I'm reading (Davis-Stober et al, 2014) contains the following statement:

A crowd is wise if a linear aggregate, for example a mean, of its members’ judgments is closer to the target value than a randomly, but not necessarily uniformly, sampled member of the crowd. (p79)

When I googled "linear aggregate" I didn't find any definitions or Wikipedia pages devoted to this concept.

It would be helpful if in addition to a definition the answerer could provide some more examples of linear aggregates, and some examples of nonlinear aggregates.

Davis-Stober, C. P., Budescu, D. V., Dana, J., & Broomell, S. B. (2014). When is a crowd wise?. Decision, 1(2), 79-101.

• I would say that is is an 'aggregate' that is defined as a linear function, e.g. a sum $x_1+x_2+\dots x_n$ or an average$\frac{x_1+x_2+\dots x_n}{n}$ are both linear aggregates, a weighted average is another example.
– user83346
Aug 30, 2015 at 13:26
• "Linear combination" is a standard term. In the context it might be intended that the coefficients sum to unity and possibly even be non-negative, which would make it a weighted arithmetic mean.
– whuber
Apr 1 at 15:36

Without getting too technical, a "linear function" is any function $$f$$ with the following two properties: $$f(x + y) = f(x) + f(y) \\ f(ax) = af(x)$$ There's a lot more detail on Wikipedia about what this definition implies.
For our purposes, it encompasses all functions that aggregate data points additively. The mean, $$\frac{\sum x}{n}$$, is probably the most famous and most useful such function. It shouldn't be hard for you to convince yourself that it is in fact a linear function (with respect to each data point) as per the above definition.
• Hello. I took the liberty of submitting an edit and writing $\frac{\sum x}{n}$ instead of $\frac{n}{\sum x}$, since that looked like a typo.