To see part 5, click here.
For the results as presented in this study, each of the grids were analyzed and then doublechecked for accuracy. We ended up with, after several weeks, 740 grids that were individually tallied and grouped according to how clustered the points were. The groupings would gather grids that had fifty percent or more of the data contained in a smaller, rectangular area of the grid of area 2125 square units, 1620 square units, 1115 square units, 610 square units and 15 square units. Any grids not falling into these categories were grouped together and deemed as not meeting the criteria. Of the 740 grids, an astonishing 90 percent of them met the criteria that fifty percent or more of the data was contained in an area less than or equal to twenty five percent of the total area of the grid, which was ten percentage points higher than the initial conjecture. The full details of the grouping in this manner are as follows:
Area of rectangle containing 50% or more of the data, square units  Number of grids meeting the criteria out of 740 / percent of total 
<25 
669 / 90.4% 
<20 
514 / 69.5% 
<15 
289 / 39.1% 
<10 
104 / 14.1% 
<5 
2 / 0.3% 
Not applicable 
71 / 9.6% 
In the same way, the totals within each category detailed above were noted, and found to be:
Area of rectangle containing 50% or more of the data, square units  Number of grids meeting the criteria out of 740 / percent of total 
21 – 25 
155 / 20.9% 
16 – 20 
225 / 30.4% 
11 – 15 
185 / 25.0% 
6 – 10 
102 / 13.8% 
1 – 5 
2 / 0.3% 
0 
71 / 9.6% 
The most interesting part is not that such a large number of the grids were clustered—rather, that such a disproportionately large number were clustered in such a small area. I think instinct would guide most people to hypothesize that the vast majority of the clusters, no matter their total quantity, would be larger clusters and not smaller ones. In other words, I think the expectation would be that a majority of the clusters would be in the 2125 square unit range, with slightly less in the 1620 square unit range, and less for smaller areas. Contrary to that belief, over sixty percent of the grids that met the criteria were in the 1120 square unit range!
To test the theory of how clustered one may perceive randomness to be in this scenario, thirtyone students were given a sheet of six blank grids and were told to “imagine” they were rolling 10side dice, much like you were earlier asked to imagine you were flipping a coin. The results of the 186 grids obtained from this exercise are as follows:
Area of rectangle containing 50% or more of the data, square units  Number of grids meeting the criteria out of 186 / percent of total 
<25 
96 / 51.6% 
<20 
38 / 20.4% 
<15 
8 / 4.3% 
<10 
0 / 0% 
<5 
0 / 0% 
Not applicable 
90 / 48.4%

Area of rectangle containing 50% or more of the data, square units  Number of grids meeting the criteria out of 186 / percent of total 
21 – 25 
58 / 31.2% 
16 – 20 
30 / 16.1% 
11 – 15 
8 / 4.3% 
6 – 10 
0 / 0% 
1 – 5 
0 / 0% 
0 
90 / 48.4% 
Here, the results imagined by the students were grossly different than from the actual random numbers, but it is clear to see that randomness is not really considered “clumpy” as we have found it to be.
So, that’s it–that’s the whole article. Admittedly, I didn’t spend enough time on the conclusion; it was a bit lackluster. I hope you’ve enjoyed the article, it was a lot of work and the activities stretched over several months. I’ll entertain a follow up entry, perhaps titled “Thoughts on The Grid,” but I’ll save that for another day.
Thanks for reading as always. Be well, and stay tuned!