I have image data, which I can represent as one dimensional vectors. Each value represents the brightness of a pixel in a line.
(1, 12, 4, 3, 1, 4)
I want to describe the distribution of brightness levels in the vector with single digit summary statistics: how far and in which direction the "bright part" of the image is away from the centre, and how broad the "bright part" is. Together they will describe a region of interest in the underlying image. In general I expect the bright area to be a unimodal peak, or if not, then a diffuse plateau, so the description of these properties doesn't need to be complex.
The vectors will be of an unknown length, so a normalised range of values would be ideal as an output.
I thought of skewness and kurtosis as they seem conceptually similar (at least graphically), but as Nick said below these have no spatial or ordered component. I thought I could use the vector indices to transform my data, and apply a "spatial skewness/kurtosis", but I can't work out how to do this.
I would prefer to use something that seems familiar to my audience, which will largely be medical professionals who have varying amounts of statistical knowledge. In this way, a "spatial skewness" would be a recognisable idea, where another method of describing things may not be.
I am not wedded to this method however (it may not even be possible). If there is a good approach that won't be too computationally intensive I am open to that too.
additional details and questions follows Nick Cox's answer below about weighted mean positions.
OK, so having thought about this a bit more I think I can try to explain more clearly my purpose. I am trying to efficiently describe the spatial distribution of values in a vector to supply to a machine learning model. The actual vectors in my data are hundreds of numbers long, and contain a lot of noise, so I am essentially trying to smooth and perform dimensionality reduction by describing the overall distribution succinctly.
The weighted mean position is great because it describes where the "mass" of values is in the vector, but it obviously doesn't describe the distribution in other ways. In particular, the shape of the distribution is missed, whether the distribution is peaked or plateau, unimodal or multimodal etc.
In an effort to not overthink things, the solution that comes to me is to also do some binned averages ie average of first 25% of vector, average of second 25% etc.
The only issue I have with this is the inelegance of it. I am using multiple values to describe one "idea", and if I want an even more fine-grained description, I add more values by making the bins smaller. This is why I had gravitated to the concept of kurtosis initially, because it describes something about shape in a single number. I realise it is not applicable at all, but I still wonder if there is a more elegant solution to describing the shape of my distributions.
I understand this is what Nick has hinted at below, with "other weighted moment measures". I am again just running against my lack of mathematical intuition. I am not sure how to make a weighted second, third or fourth moment, let alone have a grasp of which one might be useful here.