Math Madness for this Madness in March [UPDATED]
UPDATE 3/26/2021: Scroll to the bottom and see new numbers now that we’re to the Sweet 16.
Math Madness for Match Madness during ….That Other Madness in March
ScoreSide currently has active campaigns running for every team playing in that men's 2021 college hoops tournament (Download the app and join a campaign here). Each college tourney campaign is set up the same way: choose an amount to donate per point scored by your favorite team during the tournament and support Feeding America (a great cause to support now more than ever).
How much should I donate per point?
You know your team, you know the cause, the only question is: what’s a “good” amount to choose per point? Too low and you get doubly disappointed as your team is booted early and you only donated a few bucks. Too high and a big run in the tournament gets you not just fun-anxious but anxious-anxious about your donation. [note: ScoreSide has you covered in that case: you can always adjust your donation after the game for up to 3 days before it gets paid out, so if things go absolutely nuts, you can fix it].
Given the random and crazy nature of the tournament, you never know what might happen and you can't guarantee anything. But, if you wanted to send a guaranteed donation, you'd just write a check. Giving with ScoreSide is about bumping up that fun-anxious right next to your thrill-anxious as your team does better and better and your donation grows.
One way to think about how to figure out that per/point donation is to find the target amount you'd like to donate. It’s the amount you’d be willing to donate if your team does well enough but not great. You saw some good basketball and you gave to a great cause and had a successful tournament all around but maybe not the one you’ll talk about for years. What’s *that* donation amount? For the sake of experiment, we’ll be using $50 as our target.
To get to what we should donate per point, the first thing we'll need to do, duh, is make an estimate on what our team might score. And that starts with what’s an average amount of points per game.
Estimating Points
There are many, many different considerations in trying to estimate points. We're going to keep it very simple and use a very rough estimate put together by Business Insider (The article is about the "perfect bracket" but suits our purposes here). It states that the "average" NCAA finals game score (which we'll use to estimate all the game scores) is 77-68. Very rough, but it will do for now.
So, how on earth can we estimate how many points our team will score in a single-elimination tournament where the number of games played is unknown?
Expected Value
(no, you don’t need to know math)
We'll focus on the idea of Expected Value. The quick and dirty definition of Expected Value is not actually ‘expected’ but more like "if I ran this random thing a gazillion times what would the average result be" and the reason estimators use that value is not that anyone actually expects that value to show up, it's just the value that has the same amount of mass above it as below it, so if your estimate is off it's basically as likely to be too high as too low.
We're trying to figure out the right amount to pledge per point to hit our goal, right? Our goal again: $50. We know it's very very unlikely we'll donate exactly $50 but we're going to see how close we can get by choosing a good donation amount per point that gets us there. What's the "best" number to choose? Well, if we controlled the universe, the best way to choose is to run the tournament a million times and see what the average number of total points our team scores and divide that into $50 and we'll have our best estimate.
Fun with Math and Probs
We can’t do that in real life, but we can do that with some math and some data in the form of probabilities. Analyzing the bracket is no new thing and there are plenty of great researchers and odds-makers that do a lot of work every year estimating the odds for teams and games and results.
We are using a great set of data from Bart Torvik to pull our round-by-round cumulative odds of each team progressing past each round.
So, we can have a good idea of the odds of each team and we have an estimate that when a team wins it scores 77 points and when it loses it scores 68. Now we just have to either run the tournament a million times (this is actually sometimes a valid approach and it's called the Monte Carlo method) or we can do some math with Expected Values and probabilities and hopefully get a good answer.
Now we just need to pick a team as an example. Since some people have in-laws in San Diego (it's true) we choose San Diego State.
Here's what Torvik provides as the probability for San Diego State getting through and winning each round:
| R64 | R32 | S16 | E8 | F4 | Champs |
|---|---|---|---|---|---|
| 68.2% | 40.3% | 15.2% | 7.1% | 3.3% | 1.1% |
Explanation: There's a 68.2% chance that San Diego will win in Round 1 and there's a 15.2% chance they'll get to the Sweet 16 and win and so on until there's a 1.1% chance of making it to the finals and winning. We use those numbers to calculate another win likelihood on a game-by-game basis which we can call the "Implied Game Win %" which you can think of as saying "If San Diego gets to this round, what are the odds they'll win that game". We need that number to estimate what SD State is likely to score in that game.
| R64 | R32 | S16 | E8 | F4 | Champs | |
|---|---|---|---|---|---|---|
| Implied Game % | 68.2% | 59.091% | 37.717% | 46.711% | 46.479% | 33.333% |
We can then make an estimate for each game (assuming they make it to that round) of the points they'll score. We do that by simply saying the "expected points" are the (win %) * 77 + (lose %) * 68. In other words, SD State have a "win %" chance of scoring 77 and a "lose %" chance of scoring 68. With our "Implied Game %" we can get a good estimate of points scored in each round.
| R64 | R32 | S16 | E8 | F4 | Champs | |
|---|---|---|---|---|---|---|
| Expected Pts | 74.138 | 73.318 | 71.395 | 72.204 | 72.183 | 71.000 |
We can't just add up those "estimated pts" though since they are estimates based on the assumption that the team got that far. So, we have to consider the overall likelihood that San Diego State gets to each round. So, for each round we can weight the expected points by the likelihood of the team even getting into that game and only then we can just add them up.
| R64 | R32 | S16 | E8 | F4 | Champs | |
|---|---|---|---|---|---|---|
| Cum. % Exp Pts | 74.138 | 50.003 | 28.772 | 10.975 | 5.125 | 2.343 |
Let's put that all that together in one:
| San Diego | R64 | R32 | S16 | E8 | F4 | Champs | Cumulative % | 68.2% | 40.3% | 15.2% | 7.1% | 3.3% | 1.1% |
|---|---|---|---|---|---|---|
| Implied Game % | 68.2% | 59.091% | 37.717% | 46.711% | 46.479% | 33.333% |
| Expected Pts | 74.138 | 73.318 | 71.395 | 72.204 | 72.183 | 71.000 |
| Cum. % Exp Pts | 74.138 | 50.003 | 28.772 | 10.975 | 5.125 | 2.343 |
Cool, a table, but what’s it mean
Think about each row in terms of “running a gazillion tournaments and calculating averages” mindset. In 1.1% of the gazillion runs, you'd see San Diego State winning the final. They'd show up in the finals 3.3% of the time (if you win your final 4 game, you are in the finals). Of those 3.3% of the games, you'd then see SD State win 1/3 of them. If you averaged all the points scored by SD State in the finals they showed up in, it would be an average score of 71.00.
The Cum. % Exp Pts row might be the oddest one to think about. The data isn't saying that SD State scores an average of 2.343 points in the final or anything like that. It's more like how many points on an average run come from the finals. Most of the time (96.7% of the time they don't make the finals) it's 0 and the other 3.3% of the time they average 71 pts, so the average number of points that come from a Finals game is 2.343.
That’s basically it. We've got the data we need to make our estimate on "What should I donate per point for San Diego State if I'd like to choose the number that is most likely to get to a $50 total".
Putting it all together
The average number of points we can expect (remember, average) is 171.36 per run. Since our goal is $50, we divide 171.36 into 50 and we get $0.29 per point. That might feel high, but even for a good team like SD State, the odds of a long run are, well, long.
| Summary Data | |
|---|---|
| Total Expected Pts | 171.36 |
| Goal Donation | $50.00 |
| Per Pt | $0.29 |
Now for the fun part of donating. At $0.29 per point, the average might be $50 over a jillion runs, but odds are that's not what we'll donate. Average is rare afterall. So, what should we be ready for?
If SD State goes out in the first game the likely average donation for that game they lose is 0.29 * 68 or $19.84. What will we donate if our team wins it all (always scoring the exact average of 77 pts)? That's about $134.81
| Summary Data | |
|---|---|
| First Round Loss Donation | $19.84 |
| All the Way Donation | $134.81 |
Choose $0.29 per point and your "expected donation" is $50.00 but you could donate anywhere between low double digits to low triple digits unless things go very very mad. Talk about fun-anxious! AND, think about how flippin' awesome you'll feel when San Diego State drains that 77th point in the finals and wins the championship!*
*And, as we said, you are always in control of your donations and can change them (up or down) for at least 3 days after each game before they are finalized and paid.
Another Comparison (taunt?) tool
We can add one more method of comparing teams now. Beyond ranking, seeds, records, legacy, you can now brag when your team has a lower “Per Point” than another team and if some jerk friend of yours is taunting your choice but your team and theirs both have the same “Per Point” then they’re no better than you. Tell ‘em to stuff it.
Here’s how the teams break down with a $50 tournament target per team:
| Team | Exp Pts | Per Point | Out in 1 | Champs |
|---|---|---|---|---|
| Gonzaga Bulldogs | 351.4 | $0.14 | $9.68 | $65.75 |
| Houston Cougars | 300.5 | $0.17 | $11.31 | $76.87 |
| Baylor Bears | 269.5 | $0.19 | $12.62 | $85.72 |
| Illinois Fighting Illini | 257.0 | $0.19 | $13.23 | $89.90 |
| Michigan State Spartans | 265.5 | $0.19 | $12.80 | $87.00 |
| Michigan Wolverines | 265.5 | $0.19 | $12.80 | $87.00 |
| Iowa Hawkeyes | 250.2 | $0.20 | $13.59 | $92.34 |
| Ohio State Buckeyes | 248.1 | $0.20 | $13.71 | $93.12 |
| Alabama Crimson Tide | 227.2 | $0.22 | $14.97 | $101.68 |
| Florida State Seminoles | 214.1 | $0.23 | $15.88 | $107.88 |
| Tennessee Volunteers | 190.5 | $0.26 | $17.85 | $121.24 |
| Texas Longhorns | 190.0 | $0.26 | $17.89 | $121.55 |
| Kansas Jayhawks | 182.6 | $0.27 | $18.62 | $126.49 |
| Villanova Wildcats | 189.5 | $0.26 | $17.94 | $121.89 |
| West Virginia Mountaineers | 189.3 | $0.26 | $17.96 | $122.00 |
| Arkansas Razorbacks | 175.3 | $0.29 | $19.39 | $131.76 |
| BYU Cougars | 164.4 | $0.30 | $20.68 | $140.53 |
| Creighton Bluejays | 168.0 | $0.30 | $20.24 | $137.50 |
| Purdue Boilermakers | 173.5 | $0.29 | $19.59 | $133.11 |
| USC Trojans | 177.5 | $0.28 | $19.16 | $130.15 |
| Oklahoma State Cowboys | 167.3 | $0.30 | $20.33 | $138.09 |
| San Diego State Aztecs | 171.4 | $0.29 | $19.84 | $134.81 |
| Virginia Cavaliers | 170.3 | $0.29 | $19.97 | $135.68 |
| Virginia Commonwealth Rams | 170.3 | $0.29 | $19.97 | $135.68 |
| Virginia Tech Hokies | 170.3 | $0.29 | $19.97 | $135.68 |
| Connecticut Huskies | 160.7 | $0.31 | $21.15 | $143.72 |
| Texas Tech Red Raiders | 163.3 | $0.31 | $20.82 | $141.45 |
| Colorado Buffaloes | 153.0 | $0.33 | $22.22 | $151.00 |
| Colorado State Rams | 153.0 | $0.33 | $22.22 | $151.00 |
| Loyola Chicago Ramblers | 146.2 | $0.34 | $23.25 | $157.97 |
| Wisconsin Badgers | 148.4 | $0.34 | $22.91 | $155.67 |
| Florida Gators | 137.9 | $0.36 | $24.66 | $167.56 |
| LSU Tigers | 125.2 | $0.40 | $27.15 | $184.46 |
| St. Bonaventure Bonnies | 125.9 | $0.40 | $27.00 | $183.45 |
| Rutgers Scarlet Knights | 119.7 | $0.42 | $28.41 | $193.02 |
| Clemson Tigers | 115.9 | $0.43 | $29.33 | $199.30 |
| Georgia Tech Yellow Jackets | 117.1 | $0.43 | $29.02 | $197.19 |
| Oregon Ducks | 117.6 | $0.43 | $28.91 | $196.43 |
| Utah State Aggies | 118.2 | $0.42 | $28.76 | $195.41 |
| Maryland Terrapins | 114.8 | $0.44 | $29.63 | $201.29 |
| Missouri Tigers | 114.1 | $0.44 | $29.81 | $202.53 |
| Georgetown Hoyas | 111.4 | $0.45 | $30.53 | $207.44 |
| North Carolina State Wolfpack | 113.7 | $0.44 | $29.91 | $203.20 |
| North Carolina Tar Heels | 113.7 | $0.44 | $29.91 | $203.20 |
| Oklahoma Sooners | 115.0 | $0.43 | $29.57 | $200.91 |
| Syracuse Orange | 105.8 | $0.47 | $32.13 | $218.30 |
| Colgate Raiders | 96.4 | $0.52 | $35.26 | $239.59 |
| North Texas Mean Green | 95.4 | $0.52 | $35.63 | $242.11 |
| Ohio Bobcats | 95.6 | $0.52 | $35.58 | $241.72 |
| UC Santa Barbara Gauchos | 94.6 | $0.53 | $35.95 | $244.27 |
| Liberty Flames | 92.5 | $0.54 | $36.76 | $249.77 |
| Oregon State Beavers | 91.2 | $0.55 | $37.29 | $253.35 |
| UCLA Bruins | 91.6 | $0.55 | $37.10 | $252.08 |
| Abilene Christian Wildcats | 87.5 | $0.57 | $38.87 | $264.05 |
| Eastern Washington Eagles | 87.0 | $0.57 | $39.07 | $265.46 |
| Winthrop Eagles | 87.3 | $0.57 | $38.93 | $264.52 |
| Wichita State Shockers | 85.1 | $0.59 | $39.96 | $271.46 |
| Drake Bulldogs | 79.9 | $0.63 | $42.58 | $289.26 |
| Grand Canyon Antelopes | 79.1 | $0.63 | $42.99 | $292.08 |
| Morehead State Eagles | 79.0 | $0.63 | $43.03 | $292.36 |
| UNC Greensboro Spartans | 78.7 | $0.64 | $43.20 | $293.51 |
| Iona Gaels | 74.8 | $0.67 | $45.47 | $308.92 |
| Drexel Dragons | 73.6 | $0.68 | $46.18 | $313.77 |
| Oral Roberts Golden Eagles | 73.6 | $0.68 | $46.18 | $313.77 |
| Cleveland State Vikings | 71.1 | $0.70 | $47.83 | $324.98 |
| Hartford Hawks | 70.9 | $0.70 | $47.94 | $325.69 |
| Mount St. Mary's Mountaineers | 69.2 | $0.72 | $49.11 | $333.66 |
| Norfolk State Spartans | 68.5 | $0.73 | $49.66 | $337.41 |
| Texas Southern Tigers | 68.9 | $0.73 | $49.33 | $335.15 |
UPDATE 3/26/2021: The Sweet 16
Now that we’re at the Sweet 16, we’ve got a couple of tables to look at. First is for anyone joining now. Given the probabilities, what is the right amount to hit our $50 goal in just the S16. The columns of data for each team are:
Team: (obvious)
Exp Pts: The total number of points projected for the remainder of the tournament
Per Point: What amount to choose per point if your goal is to donate $50 for the rest of the tournament for your team
Out-in-1: How much would you be expected to donate (at the Per Point amount) if they lose their next game
Original: What was the original amount projected to hit the $50 goal
Current Total: what you have donated so far if you joined at the original amount
| Team | Exp Pts | Per Point | Out-in-1 | Champs | Original | Current Tot. |
|---|---|---|---|---|---|---|
| Gonzaga Bulldogs | 226.2 | $0.22 | $15.03 | $102.12 | $0.14 | $25.90 |
| Houston Cougars | 197.6 | $0.25 | $17.21 | $116.91 | $0.17 | $25.50 |
| Baylor Bears | 177.3 | $0.28 | $19.18 | $130.31 | $0.19 | $29.45 |
| Arkansas Razorbacks | 169.2 | $0.30 | $20.09 | $136.49 | $0.29 | $44.37 |
| Loyola Chicago Ramblers | 160.4 | $0.31 | $21.20 | $144.02 | $0.34 | $48.28 |
| Alabama Crimson Tide | 149.7 | $0.33 | $22.72 | $154.33 | $0.22 | $36.08 |
| Michigan Wolverines | 146.9 | $0.34 | $23.14 | $157.24 | $0.19 | $31.92 |
| USC Trojans | 142.8 | $0.35 | $23.81 | $161.75 | $0.28 | $43.96 |
| Florida State Seminoles | 128.1 | $0.39 | $26.54 | $180.34 | $0.23 | $31.05 |
| Villanova Wildcats | 115.7 | $0.43 | $29.40 | $199.73 | $0.26 | $40.82 |
| UCLA Bruins | 106.1 | $0.47 | $32.04 | $217.70 | $0.55 | $124.30 |
| Oregon Ducks | 100.8 | $0.50 | $33.73 | $229.19 | $0.43 | $40.85 |
| Oregon State Beavers | 93.7 | $0.53 | $36.27 | $246.44 | $0.55 | $82.50 |
| Syracuse Orange | 94.5 | $0.53 | $35.99 | $244.51 | $0.47 | $71.91 |
| Creighton Bluejays | 87.9 | $0.57 | $38.69 | $262.86 | $0.30 | $40.50 |
| Oral Roberts Golden Eagles | 78.3 | $0.64 | $43.41 | $294.95 | $0.68 | $106.08 |
How the other 48 Fared
For the unfortunate many that are done playing, how did donors that chose the target amount above fare? The results: average donation: $39.38. Standard Deviation: $17.91
Data are:
Team: (obvious)
Donation Per Pt: What we calculated above as the amount that would give you the best chance at donating $50 in the tournament
Points Scored: the actual number of points your team scored before they were ousted
Donation: what you actually donated with the recommended amount
| Team | Donation Per Pt. | Points Scored | Donation |
|---|---|---|---|
| Texas Longhorns | $0.26 | 52 | $13.52 |
| Tennessee Volunteers | $0.26 | 56 | $14.56 |
| Michigan State Spartans | $0.19 | 80 | $15.20 |
| Connecticut Huskies | $0.31 | 54 | $16.74 |
| Virginia Cavaliers | $0.29 | 58 | $16.82 |
| San Diego State Aztecs | $0.29 | 62 | $17.98 |
| BYU Cougars | $0.30 | 62 | $18.60 |
| Purdue Boilermakers | $0.29 | 69 | $20.01 |
| Virginia Tech Hokies | $0.29 | 70 | $20.30 |
| Utah State Aggies | $0.42 | 53 | $22.26 |
| Clemson Tigers | $0.43 | 56 | $24.08 |
| St. Bonaventure Bonnies | $0.40 | 61 | $24.40 |
| Georgia Tech Yellow Jackets | $0.43 | 60 | $25.80 |
| Illinois Fighting Illini | $0.19 | 136 | $25.84 |
| North Carolina Tar Heels | $0.44 | 62 | $27.28 |
| Missouri Tigers | $0.44 | 68 | $29.92 |
| Wichita State Shockers | $0.59 | 52 | $30.68 |
| Liberty Flames | $0.54 | 60 | $32.40 |
| Georgetown Hoyas | $0.45 | 73 | $32.85 |
| UC Santa Barbara Gauchos | $0.53 | 62 | $32.86 |
| Iowa Hawkeyes | $0.20 | 166 | $33.20 |
| Drexel Dragons | $0.68 | 49 | $33.32 |
| UNC Greensboro Spartans | $0.64 | 54 | $34.56 |
| Colgate Raiders | $0.52 | 68 | $35.36 |
| Winthrop Eagles | $0.57 | 63 | $35.91 |
| Iona Gaels | $0.67 | 55 | $36.85 |
| Ohio State Buckeyes | $0.52 | 72 | $37.44 |
| Mount St. Mary's Mountaineers | $0.72 | 52 | $37.44 |
| Hartford Hawks | $0.70 | 55 | $38.50 |
| Kansas Jayhawks | $0.27 | 144 | $38.88 |
| Cleveland State Vikings | $0.70 | 56 | $39.20 |
| West Virginia Mountaineers | $0.26 | 156 | $40.56 |
| Texas Tech Red Raiders | $0.31 | 131 | $40.61 |
| Morehead State Eagles | $0.63 | 67 | $42.21 |
| Colorado State Rams | $0.33 | 140 | $46.20 |
| Grand Canyon Antelopes | $0.63 | 74 | $46.62 |
| Eastern Washington Eagles | $0.57 | 84 | $47.88 |
| Colorado Buffaloes | $0.33 | 149 | $49.17 |
| Wisconsin Badgers | $0.34 | 148 | $50.32 |
| Rutgers Scarlet Knights | $0.42 | 120 | $50.40 |
| Florida Gators | $0.36 | 153 | $55.08 |
| Abilene Christian Wildcats | $0.57 | 100 | $57.00 |
| Oklahoma State Cowboys | $0.43 | 139 | $59.77 |
| Oklahoma Sooners | $0.43 | 143 | $61.49 |
| LSU Tigers | $0.40 | 154 | $61.60 |
| Maryland Terrapins | $0.44 | 140 | $61.60 |
| Ohio Bobcats | $0.52 | 120 | $62.40 |
| Drake Bulldogs | $0.63 | 109 | $68.67 |
| North Texas Mean Green | $0.52 | 139 | $72.28 |
| Norfolk State Spartans | $0.73 | 109 | $79.57 |
| Texas Southern Tigers | $0.73 | 126 | $91.98 |