Is there any metric which indicates that there is a large value within a list of numbers? I don't even know if this is the right place to ask, but I think I'll just do so.
I have a list of (rational) numbers, and I want a metric which indicates the confidence of "There is ONE large number compared to the others". This metric should not be effected by the scale of values, for example:

*

*1 1 100 1 1

*100 10000 100 100 100
Both of these should get somehow similar values using this metric.
Another example:


*-100 -100 -1 -100 -100
This -1 is also considered as "having a large value compared to others".
Is there any existing metric that I could use to derive such value? Or how could I construct one  myself?
I would apologize first if this is a rather naive question. I'm personally CS major instead of statistics, and I'm trying to find something to use for my neural network loss function.
Thanks in advance!
 A: This is related to the concept of "outlier detection", which has a number of implicit assumptions, not always clearly stated. Here is an example of two methods for outlier detection based on different summary statistics:
Robust Outlier Detection.
A peculiarity of your examples is that the lists of numbers only contain two values. this implies that the NMAD (according to the notation in the previous link) is zero. In such case (see)

A scaling factor of zero means the median deviation is zero and any value that is not the median of the population will be returned as an outlier

A: There are several ways to find these extreme relative outliers:
The easiest way to find it would be the maximum z-score of the list of numbers.
First, you transform all values to z-scores by subtracting the mean    and dividing by the standard deviation. Then, you look at the range    of values. Usually, the z-scores should range between -2 and 2. If    there is one outlier in the data, two things will happen: it will    have an unusually high z-score and it will be the only one with a    positive sign. So, you can spot it immediately.
Or you can look at the ratio of mean and median. An outlier will    increase the mean but the median will not be affected. In a normal    distribution, mean and median are equal. If there are high outliers,    the mean will be higher than the median. If you compute the ratio    between the two you will have an indicator for outliers.
If it is just for reporting purposes, you can also compute the    skewness of the distribution, which will be strongly positive if    there is one high value. It's a standard measure for deviations like    that.
A: Martin already gave a very good answer. Here is an alternative that makes explicit use of the fact that you only have a single very large value you are looking for.
Consider a vector $x$ that you want to look through. Let $m$ be a vector of the same length as $x$, which at the $i$-th position contains the "leave one out mean" of $x$, i.e., the mean of the entries of $x$ except the one at the $i$-th position. Similarly, let $s$ contain the "leave one out standard deviations".
The idea is that if you have a single large value at position $i$, then both $m$ and $s$ will have a strong dip at that position. You can thus compare $x$ with a detection limit of the form $m+\theta s$ with a tuning parameter $\theta$, or alternatively calculate $\frac{x-m}{s}$ componentwise and look for large values in this expression (and you would still need to define "large", which is equivalent to setting $\theta$).
In the examples you give, $s$ will be undefined at the position of interest, since all the other entries are identical, so you would need to treat that possibility separately if it is realistic.
