How a point spread becomes a win probability
A point spread is not a prediction that the favorite will win by exactly that much. It is the market's estimate of the middle of all the ways the game could go. Real final margins scatter around the spread in a roughly bell-shaped cloud, and in 1991 the statistician Hal Stern measured that cloud for the NFL: final margin minus closing spread is approximately normal with a standard deviation of about 13.86 points. That one number is the whole trick. Once you know the center of the curve (the spread) and its width (13.86), the favorite's chance of winning outright is simply the share of the curve that sits above zero.
Later analysts re-measuring longer samples get figures between 13.45 and 13.86 depending on the era. The difference sounds meaningful and is not: at a 7-point spread it moves the answer by about half a point of probability. We use Stern's published 13.86, because it is the peer-reviewed figure the whole modeling tradition descends from, and because pretending to more precision than the model has would miss the point of the model.
The formula
Φ is the standard normal cumulative distribution function, the familiar bell-curve lookup from every statistics course; we compute it with the Abramowitz and Stegun approximation, accurate to about seven decimal places. The spread goes in as a positive number, and the underdog's chance is simply 100% minus the favorite's. A pick em (spread of zero) gives exactly 50% by construction, which is the model being honest about what a spread of zero means: the market cannot separate the teams.
The moneyline path needs no curve at all. Each American moneyline converts to an implied probability (a -320 favorite implies 320 ÷ 420 = 76.19%), the two sides sum to more than 100 because the bookmaker's fee is baked in, and dividing each side by the sum removes the fee. For converting individual prices between American, decimal, and fractional formats, our odds calculator owns that job.
Worked example
A 7-point favorite. z = 7 ÷ 13.86 = 0.5051. Look that up on the normal curve: Φ(0.5051) = 0.6932, so the favorite wins outright 69.32% of the time and the underdog 30.68%. Nearly a third of touchdown favorites lose the game. That is not the model hedging; that is football.
The same game's moneylines, -320 and +260. The favorite's price implies 320 ÷ 420 = 76.19%; the underdog's implies 100 ÷ 360 = 27.78%. They sum to 103.97%, so 3.97 points of that market is the bookmaker's fee. Scale both back to a 100 total: the fair probabilities are 73.28% for the favorite and 26.72% for the underdog.
Notice the two answers disagree: the model says 69.3%, the devigged market says 73.3%. When that happens, trust the moneyline. The spread model is a translation built on one historical constant; the moneyline is the actual price the market put on this exact question, with this quarterback's elbow and this weather forecast already argued over.
The key numbers: why 3 and 7 bend the curve
The normal curve is smooth, and NFL margins are anything but. Field goals are worth 3 and touchdowns 7, so real final margins pile up on exactly those numbers: around one game in six lands on a margin of exactly 3, which a smooth bell curve considers absurd. The model is honestly a smooth approximation of a lumpy world, and the lumps push the truth around in a predictable direction. When a favorite wins by exactly the key number, that is still an outright win, so at spreads of 7 and beyond the smooth model runs a few points conservative against history. Just below the key numbers it leans slightly generous instead.
| Spread | Model says | History says (approx.) |
|---|---|---|
| 1 | 52.9% | 51 to 53% |
| 2.5 | 57.2% | 54 to 56% |
| 3 | 58.6% | 58 to 60% |
| 6.5 | 68.0% | 71 to 73% |
| 7 | 69.3% | 72 to 75% |
| 10 | 76.5% | 78 to 82% |
| 14 | 84.4% | 88 to 92% |
The history column is deliberately a range: games at any exact spread are a small sample, and different data sets disagree by a couple of points. The honest summary is that the model and history agree closely around a field goal, and the model undersells big favorites by three to six points because blowouts (and wins by exactly 7) are more common than a thin normal tail expects. One more wrinkle the smooth curve ignores: about one NFL game in 250 ends in a tie, a sliver the model quietly splits between the two teams.
Home field is already in the line
The single most common mistake with this model: adding something for home field advantage. Do not. Oddsmakers build roughly 1.5 to 2.5 points of home edge into the spread before it is ever posted, so a home team favored by 3 has already collected its credit for the crowd, the travel, and the familiar locker room. The line moved for you. If you add points on top, you are counting the same advantage twice, and your probability will be flattered in exactly the way that loses arguments. The same logic covers injuries, weather, and revenge narratives: if the market knows about it, it is in the number.
And the plain sentence this page owes you: it prices probabilities, it does not find edges. Every posted price includes the bookmaker's fee, which makes the expected value of a bet negative by construction, and everything this model knows is public information the market priced before breakfast. Use it to understand what a line is actually claiming, to check a hot take against arithmetic, and to win the argument at the watch party. That last one it is genuinely good at.