Given a real-valued data set $ x\_1, \dots, x\_n $, what do you call the quantity


$$\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits\_{i=1}^n |x\_i - x|$$


This seems like a pretty basic thing to ask for. For example, in a game in which I have to guess the number you're thinking of, and I have to pay you the difference between what I guessed and what you were thinking, the optimal strategy is to guess this quantity. This game seems like a decent model for, say, predicting the stock market and buying and selling accordingly.


I realize that the quantity is not in general unique, but that doesn't mean it isn't useful. Why haven't I heard more about this quantity? Are there any applications in which it is used?



name for $\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$
The value of $x$ that satisfies this is the median of $x\_i$. It minimizes $L\_1$ loss.


Note that the mean of $x\_i$ is the value of $x$ minimizing $\sum(x\_i - x)^2$, the $L\_2$ loss.


