I have a list of ~20 numbers:
1200, 1200, 360, 360, 300, 250, 180, 180, 180, 180, 180, 90, 90, 90, 90, 45, 10, 0, 0
I am looking for a metric that determines the lopsidedness (maybe skewness) of this distribution. For the above example, I would want to be able to highlight that the sum of the first 2 numbers (2400) make up nearly 50% of the total sum (4985). But there could also be other examples where the top 4 or 5 numbers make up a big percentage (say, greater than 50%) of the total sum.
Should I just calculate skewness or are there other better metrics that fulfill my requirement?
It's not entirely clear what precisely you want your 'lopsidedness measure' to reveal. However, I found your question interesting, and so I invented a 'lopsidedness measure', $\mathcal{L}$, that you might find useful (Before I continue, I want to make clear that even though I said I invented this, almost certainly I'm not the first to think of this. I just haven't personally heard of it before.)
How to calculate $\mathcal L$:
First, make sure that your array of numbers is sorted in an decreasing order (As yours are):
Next, calculate the cumulative sum of these numbers
Now divide by the total sum of your original vector (which happens to be the last element of the cumulative sum, namely 4985):
Now take the average of these numbers. The result is $$\mathcal L = 0.810$$
Properties of $\mathcal L$:
The measure $\mathcal L$ is a real number in the interval $(0.5, 1]$, i.e. $0.5<\mathcal L \leq 1$.
When the data is at its most lopsidedness, that is when all the weight is gathered in a single entry in the sequence of numbers, then $\mathcal L = 1$.
On the other hand, the less and less lopsidedness the sequence of numbers are, i.e. the closer the weight is to being uniformly distributed, the closer the measure comes close to a half. The way to see this is the following. Create a vector of some length $n$, with all numbers equal, then you will find that $\mathcal L \approx 0.5$. If you continuously increase the numbers of elements in your sequence, if you let $n\to \infty$, then you will have that $\mathcal L \to 0.5$ from above.
Examples:
In your example, you see that your numbers are moderately 'lopsided' with $\mathcal L = 0.810$ being a bit higher than midway between $0.5$ and $1$.
Now try with a more lopsided sequence of numbers:
from which we calculate $\mathcal L = 0.946$.
Lastly, we try with a sequence of numbers that is very non-lopsided (i.e. weight close to evenly distributed):
from which we calculate $\mathcal L = 0.555$.