Instead of looking at global properties of the pattern (like symmetries), one can take a look at the local ones, e.g. the number of neighbors each stone (=black circle) has. Let's denote the total number of stones by $s$.

If the stones where thrown at random, the distribution of neighbors is $$P_{rand,p}(k\ \text{neighbors}|n\ \text{places} ) = {n \choose k} p^{k} (1-p)^{n-k},$$ where $p = s/25$ is the density of stones. Number of places $n$ depends if a stone is in the interior ($n=8$), on the edge ($n=5$) or on the corner $(n=3)$.

It is clearly visible, that distribution of neighbors in C), D) and E) are far from random. For example, for D) all interior stones have exactly $4$ neighbors (opposing to random distribution, which yields in $\approx (0\%,2\%,9\%,20\%,27\%,24\%,13\%,4\%,0\%)$ instead of the measured $(0\%,0\%,0\%,0\%,100\%,0\%,0\%,0\%,0\%)$).

So to quantify if a pattern is random you need to compare its distribution of neighbors $P_{measured}(k|n)$ and compare it with a random one $P_{rand,p}(k|n)$. For example you can compare their means and variances.

Alternatively, one can measure their distances in the function spaces, e.g.: $$\sum_{n=\{3,5,8\}} \sum_{k=0}^n\left[P_{measured}(k|n)P_{measured}(n) -P_{rand,p}(k|n)P_{rand,p}(n)\right]^2,$$ where $P_{measured}(n)$ is the measured ratio of points with $n$ adjacent spaces and $P_{rand,p}(n)$ is the predicted for random pattern, i.e. $P_{measured}(3) = 4/25$, $P_{measured}(5) = 12/25$ and $P_{measured}(8) = 9/25$.

Instead of looking at global properties of the pattern (like symmetries), one can take a look at the local ones, e.g. the number of neighbors each stone (=black circle) has. Let's denote the total number of stones by $s$.

If the stones where thrown at random, the distribution of neighbors is $$P_{rand,p}(k\ \text{neighbors}|n\ \text{places} ) = {n \choose k} p^{k} (1-p)^{n-k},$$ where $p = s/25$ is the density of stones. Number of places $n$ depends if a stone is in the interior ($n=8$), on the edge ($n=5$) or on the corner $(n=3)$.

It is clearly visible, that distribution of neighbors in C), D) and E) are far from random. For example, for D) all interior stones have exactly $4$ neighbors (opposing to random distribution, which yields in $\approx (0\%,2\%,9\%,20\%,27\%,24\%,13\%,4\%,0\%)$ instead of the measured $(0\%,0\%,0\%,0\%,100\%,0\%,0\%,0\%,0\%)$).

So to quantify if a pattern is random you need to compare its distribution of neighbors $P_{measured}(k|n)$ and compare it with a random one $P_{rand,p}(k|n)$. For example you can compare their means and variances.

Alternatively, one can measure their distances in the function spaces, e.g.: $$\sum_{n=\{3,5,8\}} \sum_{k=0}^n\left[P_{measured}(k|n)P_{measured}(n) -P_{rand,p}(k|n)P_{rand,p}(n)\right]^2,$$ where $P_{measured}(n)$ is the measured ratio of points with $n$ adjacent spaces and $P_{rand,p}(n)$ is the predicted for a random pattern, i.e. $P_{rand,p}(3) = 4/25$, $P_{rand,p}(5) = 12/25$ and $P_{rand,p}(8) = 9/25$.