I have an n x n matrix. Each cell contains a value. The matrix is essentially a heatmap. The null hypothesis is that the greatest values would be at the horizontal and vertical midlines of the matrix. Let's say that n is odd. Then, the columns of the matrix would correspond to x = -(n-1)/2 : (n-1) / 2. The rows would correspond to y = -(n-1)/2 : (n-1) / 2. And therefore the middle column and middle row would correspond to x = 0 and y = 0, respectively. The null hypothesis is that the hottest cells would lie close to x = 0 and y = 0. The alternative hypothesis is that the hottest cells are off both midlines, in other words, displaced from the origin.
I can use a weighted centroid to find the center of mass of the matrix. But what sort of significance test could I use to confirm the alternative hypothesis?
Edit: Regarding the null hypothesis. The null hypothesis is as follows. Take triples (a,b,c) <-- [1, 100]^3. In other words for each triple, a,b,c are taken from the uniform distribution on [1, 100]. then, let x = a-b, and y = b-c. So, when many random triples are obtained, x and y are assigned, and the results are plotted, you actually have a distribution that favors the top left and bottom right quadrants, and appears relatively more rarely in the bottom left and top right quadrants. An example is below.