Using Pythagorean win expectations in Wyoming high school football to find the most surprising seasons in state history

If you just look at the cumulative effect of NSI Academy’s scores from the 2007 season, you would not have expected much from the Wolves.

For the year, the Wolves were outscored 281-150.

And with a score differential like that, what kind of record would you expect?

Well, there’s some math we can do to figure that out.

Using something called the Pythagorean expectation, the Wolves — who played 10 games that season — would have been roughly expected to finish with a record of about 2-8. After all, most teams with that kind of score differential across that many games finish with about that kind of record.

But the Wolves finished 6-4. They won 4.2 more games than the Pythagorean model would have expected them to.

And across more than 100 years of Wyoming high school football, the Wolves’ 4.2 wins above expectation is the highest difference for a single team in state history.

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Here’s the math part, as simplified as I can make it: In short, the Pythagorean expectation model looks at a team’s scoring differential and then tries to estimate what a team’s win-loss record should be based on that differential alone. For example, a team that goes 4-4 and finished with a scoring differential of zero would have a Pythagorean score of 0.0 — and the logic follows that you’d expect a team that scored as many points as it gave up to have a .500 record.

A positive Pythagorean score means you won more games than the model would predict; a negative Pythagorean score means you lost more. (For a breakdown of the math used in this post, check out Wikipedia’s article on Pythagorean expectation.)

For a more recent example of the Pythagorean model, let’s look at the 2023 season, and two teams that share a both a 2023 victory total and a first letter: Worland and Wright.

Worland, playing in Class 3A, finished 3-6. Wright, a Class 1A nine-man team, finished 3-5. So we would expect their point differential to be about the same, right?

Well… Would you believe that Wright outscored its opponents 201-124, while Worland was outscored 286-69?

The Pythagorean expectations vary greatly because of those totals. Through that math, we would have expected Wright to win 6.1 games out of its eight with that kind of scoring margin. Instead, Wright won three, giving Wright a score of negative 3.1 (3 wins minus 6.1 predicted wins = negative 3.1). Meanwhile, we’d expect Worland to win 0.3 games out of its nine, so the Warriors’ Pythagorean score is a positive 2.7 (3-0.3=2.7).

In short: With their score differentials, Wright should have won about six games; Worland should have won zero. Instead, both teams won three games. Together, they were the biggest outliers of the 2023 season, Wright the team with the lowest number of victories from what would be expected from their season point differential, Worland the highest.

But they are still far from some of the biggest single-season outliers we’ve ever seen.

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Across the 5,469 Wyoming high school football seasons where a team played at least five games, the Pythagorean expectations correctly predicted a team’s record within one game in 3,596 cases, or about 66% of the time. (Yes, I ran the math for all 5,469.)

The outliers, though — like NSI’s 2007 season — are the most interesting.

NSI’s 2007 season was by far the biggest outlier on the positive side. On the negative side, well… meet Rawlins, whose 1928 team had a Pythagorean win expectation of 7.3 wins over their eight games. The Outlaws finished the season with a 151-56 point differential. Still, Rawlins only mustered a 3-3-2 record, a Pythagorean win expectation that was -4.3 fewer than expected. It’s the lowest mark in state history.

Oddly enough, just two years later, basically the same thing happened to Rawlins, which tied with Green River’s 1940 team for the second-worst Pythagorean outcome in state history. That year, Rawlins outscored opponents 112-41 and had a Pythagorean win expectation of 8.2 across nine games. The Outlaws went 4-4-1, a -4.2 below expectations.

The Pythagorean expectations can also be extrapolated to predict winning percentage, as well. Below are some of the best and worst outcomes we’ve seen using the Pythagorean expectations model for teams that played a minimum of five games in their season:

Best Pythagorean win differences (overachievers)
1. NSI 2007 (+4.2): Predicted 1.8 victories, actual record 6-4, point differential 150-281
2. Gebo 1932 (+3.6): Predicted 1.4 victories, actual record 5-3, point differential 66-126
3. Cheyenne East 2008 (+3.4): Predicted 3.6 victories, actual record 7-3, point differential 207-263
4t. Glenrock 1995 (+3.3): Predicted 1.7 victories, actual record 5-4, point differential 85-157
4t. Greybull 1991 (+3.3): Predicted 1.7 victories, actual record 5-3, point differential 93-161
4t. Kemmerer 1981 (+3.3): Predicted 1.7 victories, actual record 5-4, point differential 80-146
7t. Cody 1926 (+3.2): Predicted 1.8 victories, actual record 5-2, point differential 72-113
7t. Natrona 1945 (+3.2): Predicted 1.8 victories, actual record 5-4, point differential 75-134
7t. Big Piney 2007 (+3.2): Predicted 2.8 victories, actual record 6-3, point differential 122-169
10t. Evanston 2021 (+3.1): Predicted 1.9 victories, actual record 5-4, point differential 135-238
10t. Torrington 1980 (+3.1): Predicted 0.9 victories, actual record 4-3, point differential 42-94
10t. Lovell 1976 (+3.1): Predicted 0.9 victories, actual record 4-5, point differential 64-161
13t. Midwest 1956 (+3.0): Predicted 1.0 victories, actual record 4-5, point differential 65-160
13t. Cody 1953 (+3.0): Predicted 1.0 victories, actual record 4-4, point differential 113-261
13t. Pine Bluffs 1975 (+3.0): Predicted 1.0 victories, actual record 4-4, point differential 65-147
16t. Rawlins 1931 (+2.9): Predicted 1.1 victories, actual record 4-3-1, point differential 47-103
16t. Wright 2005 (+2.9): Predicted 4.1 victories, actual record 7-4, point differential 123-153
18t. Ten Sleep 1978 (+2.8): Predicted 1.2 victories, actual record 4-4, point differential 66-139
18t. Cody 1978 (+2.8): Predicted 1.2 victories, actual record 4-5, point differential 60-132
18t. Star Valley 1974 (+2.8): Predicted 1.2 victories, actual record 4-4, point differential 79-162
18t. Superior 1955 (+2.8): Predicted 1.2 victories, actual record 4-4, point differential 84-172
18t. Lingle 1925 (+2.8): Predicted 0.2 victories, actual record 3-7, point differential 58-274

Worst Pythagorean win differences (underachievers)
1. Rawlins 1928 (-4.3): Predicted 7.3 victories, actual record 3-3-2, point differential 151-56
2t. Rawlins 1930 (-4.2): Predicted 8.2 victories, actual record 4-4-1, point differential 112-41
2t. Green River 1940 (-4.2): Predicted 7.2 victories, actual record 3-2-3, point differential 94-37
4. Cheyenne Central 1920 (-4.0): Predicted 8.0 victories, actual record 4-3-2, point differential 98-40
5t. Torrington 1925 (-3.9): Predicted 6.9 victories, actual record 3-4-1, point differential 159-72
5t. Green River 1965 (-3.9): Predicted 5.9 victories, actual record 2-3-3, point differential 115-74
5t. Lingle 1936 (-3.9): Predicted 7.9 victories, actual record 4-2-3, point differential 71-31
8. Torrington 1922 (-3.8): Predicted 7.8 victories, actual record 4-5, point differential 222-101
9t. Kemmerer 1956 (-3.7): Predicted 7.7 victories, actual record 4-4-1, point differential 149-70
9t. Green River 1929 (-3.7): Predicted 6.7 victories, actual record 3-2-2, point differential 72-20
11t. Sheridan 1954 (-3.6): Predicted 7.6 victories, actual record 4-3-2, point differential 168-81
11t. Natrona 1936 (-3.6): Predicted 10.6 victories, actual record 7-1-3, point differential 156-40
11t. Kemmerer 1931 (-3.6): Predicted 9.6 victories, actual record 6-2-2, point differential 169-46
11t. Rawlins 1940 (-3.6): Predicted 9.6 victories, actual record 6-3-1, point differential 191-52
Since 2000:
1. Burlington 2005 (-3.5): Predicted 8.5 victories, actual record 5-4, point differential 322-101
2. Mountain View 2000 (-3.3): Predicted 8.3 victories, actual record 5-5, point differential 253-131
3. Wright 2023 (-3.1): Predicted 6.1 victories, actual record 3-5, point differential 201-124
4t. Rock Springs 2019 (-3.0): Predicted 7.0 victories, actual record 4-6, point differential 246-172
4t. Sheridan 2020 (-3.0): Predicted 10.0 victories, actual record 7-4, point differential 390-148
6. Pine Bluffs 2011 (-2.9): Predicted 7.9 victories, actual record 5-4, point differential 196-84

Best Pythagorean win percentage differences (overachievers)
1. Superior 1944: Predicted win percentage 0.083, actual win percentage 0.600, difference +0.517, point differential 21-58
2. Cody 1926: Predicted win percentage 0.256, actual win percentage 0.714, difference +0.459, point differential 72-113
3. Gebo 1932: Predicted win percentage 0.178, actual win percentage 0.625, difference +0.447, point differential 66-126
4. Torrington 1980: Predicted win percentage 0.129, actual win percentage 0.571, difference +0.442, point differential 42-94
5. Rawlins 1931: Predicted win percentage 0.135, actual win percentage 0.563, difference +0.428, point differential 47-103
6. Lovell 1942: Predicted win percentage 0.158, actual win percentage 0.583, difference +0.425, point differential 38-77
7. NSI 2007: Predicted win percentage 0.184, actual win percentage 0.600, difference +0.416, point differential 150-281
8. Greybull 1991: Predicted win percentage 0.214, actual win percentage 0.625, difference +0.411, point differential 93-161
9. Sunrise 1949: Predicted win percentage 0.208, actual win percentage 0.600, difference +0.392, point differential 100-176
10. Guernsey 1937: Predicted win percentage 0.215, actual win percentage 0.600, difference +0.385, point differential 33-57
Since 2000:
1. NSI 2007: Predicted win percentage 0.184, actual win percentage 0.600, difference +0.416, point differential 150-281
2. Big Piney 2007: Predicted win percentage 0.316, actual win percentage 0.667, difference +0.351, point differential 122-169
3. Evanston 2021: Predicted win percentage 0.207, actual win percentage 0.556, difference +0.349, point differential 135-238
4. Cheyenne East 2008: Predicted win percentage 0.362, actual win percentage 0.700, difference +0.338, point differential 207-263
5. Saratoga 2015: Predicted win percentage 0.070, actual win percentage 0.375, difference +0.305, point differential 98-291

Worst Pythagorean win percentage differences (underachievers)
1. Sundance 1933: Predicted win percentage 0.834, actual win percentage 0.333, difference -0.500, point differential 79-40
2. Gebo 1930: Predicted win percentage 0.884, actual win percentage 0.400, difference, -0.484, point differential 59-25
3. Sunrise 1939: Predicted win percentage 0.875, actual win percentage 0.400, difference -0.475, point differential 59-26
4. Wheatland 1981: Predicted win percentage 0.816, actual win percentage 0.375, difference -0.441, point differential 133-71
5. Cowley 1925: Predicted win percentage 0.576, actual win percentage 0.143, difference -0.434, point differential 82-72
6. Lingle 1940: Predicted win percentage 0.931, actual win percentage 0.500, difference -0.431, point differential 132-44
7. Torrington 1925: Predicted win percentage 0.867, actual win percentage 0.438, difference -0.430, point differential 159-72
8. Torrington 1922: Predicted win percentage 0.866, actual win percentage 0.444, difference -0.422, point differential 222-101
9. Rawlins 1942: Predicted win percentage 0.916, actual win percentage 0.500, difference -0.416, point differential 107-39
10. Rawlins 1930: Predicted win percentage 0.915, actual win percentage 0.500, difference -0.415, point differential 112-41
Since 2000:
1t. Burlington 2005: Predicted win percentage 0.940, actual win percentage 0.556, difference -0.384, point differential 322-101
1t. Wright 2023: Predicted win percentage 0.759, actual win percentage 0.375, difference -0.384, point differential 201-124
3t. Mountain View 2000: Predicted win percentage 0.828, actual win percentage 0.500, difference -0.326, point differential 253-131
3t. Pine Bluffs 2011: Predicted win percentage 0.882, actual win percentage 0.556, difference -0.326, point differential 196-84
5. Lusk 2019: Predicted win percentage 0.803, actual win percentage 0.500, difference -0.303, point differential 304-168

–patrick