When you look up articles about what "average" means, the standard answers are mode, median, and (arithmetic) mean. Wikipedia's Average page doesn't even mention 'hypotenuse' in the article's "see also" section at the bottom. And yet, hypotenuse is another simple (Pythagorean) formula for taking two inputs and arriving at a third which scales with input.

Hypotenuse – the line of a right triangle opposite the 90-degree corner – approaches the length of the longer number as the shorter number approaches zero, which means that unlike arithmetic mean, the resulting hypotenuse number will never be less than either of the two inputs. Would this ever be useful in sorting lists of numbers? To answer this question, I wanted to see the situation visualized, so I prepared these charts.

### Arithmetic Mean

This first chart is simple – the mean of the column and row. To visualize, each cell is yellowed by the value's percentage of 100. The result is an even spread of low (unshaded) values in the upper left corner to high (yellow) values in the lower right. There are no curves in the shading – yellowing happens in straight lines.

Mean 0 10 20 30 40 50 60 70 80 90 100
0 0 5 10 15 20 25 30 35 40 45 50
10 5 10 15 20 25 30 35 40 45 50 55
20 10 15 20 25 30 35 40 45 50 55 60
30 15 20 25 30 35 40 45 50 55 60 65
40 20 25 30 35 40 45 50 55 60 65 70
50 25 30 35 40 45 50 55 60 65 70 75
60 30 35 40 45 50 55 60 65 70 75 80
70 35 40 45 50 55 60 65 70 75 80 85
80 40 45 50 55 60 65 70 75 80 85 90
90 45 50 55 60 65 70 75 80 85 90 95
100 50 55 60 65 70 75 80 85 90 95 100

### Hypotenuse

Similar to the previous chart, hypotenuse is calculated between the row and column. I then took those hypotenuse values and calculated them as percentage of the highest value (141.42...) in order to bring the scale down to the same as mean. Since I'm building this comparison for the purpose of sorting, this allows us to directly compare how hypotenuse would sort values different from arithmetic mean.

A quick glance at the chart shows similar coloring to the previous. But a closer look shows two differences: this chart is, on average, yellower. And second, there is now a curve to the shape of the shading.

Hypotenuse 0 10 20 30 40 50 60 70 80 90 100
0 0 7 14 21 28 35 43 50 57 64 71
10 7 10 16 22 29 36 43 50 57 64 71
20 14 16 20 26 32 38 45 52 58 65 72
30 21 22 26 30 35 41 48 54 61 67 74
40 28 29 32 35 40 45 51 57 63 70 76
50 35 36 38 41 45 50 55 61 67 73 79
60 43 43 45 48 51 55 60 65 71 77 83
70 50 50 52 54 57 61 65 70 75 81 87
80 57 57 58 61 63 67 71 75 80 85 91
90 64 64 65 67 70 73 77 81 85 90 95
100 71 71 72 74 76 79 83 87 91 95 100

### Difference

In order to see exactly how the arithmetic mean and hypotenuse differ in sorting, this chart shows the results of subtracting arithmetic mean from hypotenuse. It now becomes plainly evident that largest differences are those where the row value is most different (percantage wise) from the column value.

It now become easier to see that hypotenuse sorting gives additional sorting weight to items which have high scores in a single value. For a quick illustration, note that in arithmetic mean the values (50,50) are equal in sorting to (100,0), whereas in hypotenuse a (100,0) score outranks even a (70,70) score.

Difference 0 10 20 30 40 50 60 70 80 90 100
0 0 2 4 6 8 10 13 15 17 19 21
10 2 0 1 2 4 6 8 10 12 14 16
20 4 1 0 1 2 3 5 7 8 10 12
30 6 2 1 0 0 1 3 4 6 7 9
40 8 4 2 0 0 0 1 2 3 5 6
50 10 6 3 1 0 0 0 1 2 3 4
60 13 8 5 3 1 0 0 0 1 2 3
70 15 10 7 4 2 1 0 0 0 1 2
80 17 12 8 6 3 2 1 0 0 0 1
90 19 14 10 7 5 3 2 1 0 0 0
100 21 16 12 9 6 4 3 2 1 0 0

### Conclusion

Like I said earlier, this information is probably bordering on tautological, especially to math whizzes, but I'm no math whiz, just a statistics junkie. A statistics junkie who understands topics more deeply when he can see them visualized like this. I am not sure if hypotenuse sorting is ever the best sorting for information, but at least now I (and maybe you) understand better the effect it would have on the results.